prompt stringlengths 12 1.27k | context stringlengths 2.29k 64.4k | A stringlengths 1 145 ⌀ | B stringlengths 1 129 ⌀ | C stringlengths 3 138 ⌀ | D stringlengths 1 158 ⌀ | E stringlengths 1 143 ⌀ | answer stringclasses 5
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A $2.00 \mathrm{~kg}$ particle moves along an $x$ axis in one-dimensional motion while a conservative force along that axis acts on it. The potential energy $U(x)$ associated with the force is plotted in Fig. 8-10a. That is, if the particle were placed at any position between $x=0$ and $x=7.00 \mathrm{~m}$, it would ha... | If a force is conservative, it is possible to assign a numerical value for the potential at any point and conversely, when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken, contributing to the mechanical energy and the... | -3.8 | 1000 | '-45.0' | 3.0 | 20.2 | D |
A playful astronaut releases a bowling ball, of mass $m=$ $7.20 \mathrm{~kg}$, into circular orbit about Earth at an altitude $h$ of $350 \mathrm{~km}$.
What is the mechanical energy $E$ of the ball in its orbit? | Hence, mechanical energy E_\text{mechanical} of the satellite-Earth system is given by E_\text{mechanical} = U + K E_\text{mechanical} = - G \frac{M m}{r}\ + \frac{1}{2}\, m v^2 If the satellite is in circular orbit, the energy conservation equation can be further simplified into E_\text{mechanical} = - G \frac{M m}{2r... | -214 | 29.36 | 0.01961 | 132.9 | 11000 | A |
Let the disk in Figure start from rest at time $t=0$ and also let the tension in the massless cord be $6.0 \mathrm{~N}$ and the angular acceleration of the disk be $-24 \mathrm{rad} / \mathrm{s}^2$. What is its rotational kinetic energy $K$ at $t=2.5 \mathrm{~s}$ ? | The rotational energy depends on the moment of inertia for the system, I . Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Knowledge of the Euler angles as function of time t and the initial coordinates \mathbf{r}(0) determine the k... | 0.38 | 2688 | 90.0 | 135.36 | 0.18162 | C |
A food shipper pushes a wood crate of cabbage heads (total mass $m=14 \mathrm{~kg}$ ) across a concrete floor with a constant horizontal force $\vec{F}$ of magnitude $40 \mathrm{~N}$. In a straight-line displacement of magnitude $d=0.50 \mathrm{~m}$, the speed of the crate decreases from $v_0=0.60 \mathrm{~m} / \mathrm... | When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material. The equation is much simpler and can help to underst... | 2.89 | 72 | 22.2 | 0.139 | -1.0 | C |
While you are operating a Rotor (a large, vertical, rotating cylinder found in amusement parks), you spot a passenger in acute distress and decrease the angular velocity of the cylinder from $3.40 \mathrm{rad} / \mathrm{s}$ to $2.00 \mathrm{rad} / \mathrm{s}$ in $20.0 \mathrm{rev}$, at constant angular acceleration. (T... | In physics, angular acceleration refers to the time rate of change of angular velocity. Therefore, the instantaneous angular acceleration α of the particle is given by : \alpha = \frac{d}{dt} \left(\frac{v_{\perp}}{r}\right). The angular velocity satisfies equations of motion known as Euler's equations (with zero appli... | -0.0301 | 635.7 | 260.0 | -233 | 209.1 | A |
A living room has floor dimensions of $3.5 \mathrm{~m}$ and $4.2 \mathrm{~m}$ and a height of $2.4 \mathrm{~m}$.
What does the air in the room weigh when the air pressure is $1.0 \mathrm{~atm}$ ? | This can be seen by using the ideal gas law as an approximation. ==Dry air== The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure: \begin{align} \rho &= \frac{p}{R_\text{specific} T}\\\ R_\text{specific} &= \frac{R}{M} = \frac{k_{\rm B}}{m}\\\ \rho &= \fr... | 4.979 | 0.72 | 2.0 | 418 | 0.14 | D |
An astronaut whose height $h$ is $1.70 \mathrm{~m}$ floats "feet down" in an orbiting space shuttle at distance $r=6.77 \times 10^6 \mathrm{~m}$ away from the center of Earth. What is the difference between the gravitational acceleration at her feet and at her head? | It is calculated as the distance between the centre of gravity of a ship and its metacentre. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from the rotation of the Earth (but the latter is small enough to be negligible for most purposes);... | -4.37 | 157.875 | 37.9 | 2.26 | 3.29527 | A |
If the particles in a system all move together, the com moves with them-no trouble there. But what happens when they move in different directions with different accelerations? Here is an example.
The three particles in Figure are initially at rest. Each experiences an external force due to bodies outside the three-par... | And any system of particles that move under Newtonian gravitation as if they are a rigid body must do so in a central configuration. In Euler's three-body problem we assume that the two centres of attraction are stationary. Together with Euler's collinear solutions, these solutions form the central configurations for t... | 1.16 | 28 | 2283.63 | 3.00 | 0.18162 | A |
An asteroid, headed directly toward Earth, has a speed of $12 \mathrm{~km} / \mathrm{s}$ relative to the planet when the asteroid is 10 Earth radii from Earth's center. Neglecting the effects of Earth's atmosphere on the asteroid, find the asteroid's speed $v_f$ when it reaches Earth's surface. | Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the r... | 2 | -7.5 | 2.74 | 0.0384 | 1.60 | E |
The huge advantage of using the conservation of energy instead of Newton's laws of motion is that we can jump from the initial state to the final state without considering all the intermediate motion. Here is an example. In Figure, a child of mass $m$ is released from rest at the top of a water slide, at height $h=8.5 ... | The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. Classically, conservation of energy was distinct from conservation of mass. The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dy... | 0.011 | 13 | 8.8 | 4.979 | 0.132 | B |
Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance. | * If one has $1000 invested for 1 year at a 7-day SEC yield of 2%, then: :(0.02 × $1000 ) / 365 ~= $0.05479 per day. * If one has $1000 invested for 30 days at a 7-day SEC yield of 5%, then: :(0.05 × $1000 ) / 365 ~= $0.137 per day. The total compound interest generated is the final value minus the initial principal: I... | 1.41 | 7.0 | '-2.0' | 0.5 | 7.25 | E |
Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200 \mathrm{~L}$ of a dye solution with a concentration of $1 \mathrm{~g} / \mathrm{L}$. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $2 \mathrm{~L} / \mathrm{min}$,... | Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. 500px|right|thumb|The l... | 26.9 | 313 | 0.66 | 460.5 | 14 | D |
A certain vibrating system satisfies the equation $u^{\prime \prime}+\gamma u^{\prime}+u=0$. Find the value of the damping coefficient $\gamma$ for which the quasi period of the damped motion is $50 \%$ greater than the period of the corresponding undamped motion. | For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: : \zeta = \frac{c}{c_c} = \frac {\text{actual damping}} {\text{critical damping}}, where the sy... | 0.08 | 0.249 | 11.0 | 1.4907 | 2.72 | D |
Find the value of $y_0$ for which the solution of the initial value problem
$$
y^{\prime}-y=1+3 \sin t, \quad y(0)=y_0
$$
remains finite as $t \rightarrow \infty$ | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. ;Second example The solution of : y'+3y=6t+5,... | 0.7071067812 | -2.5 | 2.0 | 272.8 | 0.24995 | B |
A certain spring-mass system satisfies the initial value problem
$$
u^{\prime \prime}+\frac{1}{4} u^{\prime}+u=k g(t), \quad u(0)=0, \quad u^{\prime}(0)=0
$$
where $g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)$ and $k>0$ is a parameter.
Suppose $k=2$. Find the time $\tau$ after which $|u(t)|<0.1$ for all $t>\tau$. | The modified KdV–Burgers equation is a nonlinear partial differential equationAndrei D. Polyanin, Valentin F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, second edition, p 1041 CRC PRESS :u_t+u_{xxx}-\alpha u^2\,u_x - \beta u_{xx}=0. ==See also== *Burgers' equation *Korteweg–de Vries equation *modifi... | 5.5 | 0.9830 | 25.6773 | 131 | 399 | C |
Suppose that a sum $S_0$ is invested at an annual rate of return $r$ compounded continuously.
Determine $T$ if $r=7 \%$. | The function f(r) is shown to be accurate in the approximation of t for a small, positive interest rate when r=.08 (see derivation below). f(.08)\approx1.03949, and we therefore approximate time t as: : t=\bigg(\frac{\ln2}{r}\bigg)f(.08) \approx \frac{.72}{r} Written as a percentage: : \frac{.72}{r}=\frac{72}{100r} Thi... | 0.00024 | 1.5377 | 0.2115 | 9.90 | 52 | D |
A mass weighing $2 \mathrm{lb}$ stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, determine the position $u$ of the mass at any time $t$. Find the frequency of the motion. | The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n =... | 0.33333333 | 6.283185307 | 0.03 | 9.30 | 0.7854 | E |
If $\mathbf{x}=\left(\begin{array}{c}2 \\ 3 i \\ 1-i\end{array}\right)$ and $\mathbf{y}=\left(\begin{array}{c}-1+i \\ 2 \\ 3-i\end{array}\right)$, find $(\mathbf{y}, \mathbf{y})$. | More precisely, given two sets of variables represented as coordinate vectors and y, then each equation of the system can be written y^TA_ix=g_i, where, is an integer whose value ranges from 1 to the number of equations, each A_i is a matrix, and each g_i is a real number. There are several possible ways to compute the... | 5.828427125 | 2500 | '-21.2' | 0.54 | 16 | E |
15. Consider the initial value problem
$$
4 y^{\prime \prime}+12 y^{\prime}+9 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-4 .
$$
Determine where the solution has the value zero. | Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). A solution to an initial v... | 131 | 0.2553 | 83.81 | 3.07 | 0.4 | E |
A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. What monthly payment rate is required to pay off the loan in 3 years? | If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are d... | 38 | 258.14 | 4.49 | 0.0000092 | 61 | B |
Consider the initial value problem
$$
y^{\prime \prime}+\gamma y^{\prime}+y=k \delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0
$$
where $k$ is the magnitude of an impulse at $t=1$ and $\gamma$ is the damping coefficient (or resistance).
Let $\gamma=\frac{1}{2}$. Find the value of $k$ for which the response has a peak v... | * Determine the system steady-state gain k=A_0with k=\lim_{t\to\infty} \dfrac{y(t)}{x(t)} * Calculate r=\dfrac{t_{25}}{t_{75}} P=-18.56075\,r+\dfrac{0.57311}{r-0.20747}+4.16423 X=14.2797\,r^3-9.3891\,r^2+0.25437\,r+1.32148 * Determine the two time constants \tau_2=T_2=\dfrac{t_{75}-t_{25}}{X\,(1+1/P)} \tau_1=T_1=\dfrac... | 2.8108 | 524 | 131.0 | 0.648004372 | 0.22222222 | A |
If a series circuit has a capacitor of $C=0.8 \times 10^{-6} \mathrm{~F}$ and an inductor of $L=0.2 \mathrm{H}$, find the resistance $R$ so that the circuit is critically damped. | In the case of the series RLC circuit, the damping factor is given by :\zeta = \frac{\, R \,}{2} \sqrt{ \frac{C}{\, L \,} \,} = \frac{1}{\ 2 Q\ } ~. For the parallel circuit, the attenuation is given byNilsson and Riedel, p. 286. : \alpha = \frac{1}{\,2\,R\,C\,} and the damping factor is consequently :\zeta = \frac{1}{... | 1000 | 4.4 | 0.264 | 25.6773 | 0.132 | A |
If $y_1$ and $y_2$ are a fundamental set of solutions of $t y^{\prime \prime}+2 y^{\prime}+t e^t y=0$ and if $W\left(y_1, y_2\right)(1)=2$, find the value of $W\left(y_1, y_2\right)(5)$. | In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{... | 3.0 | 30 | 5.41 | 0.08 | 32 | D |
Consider the initial value problem
$$
5 u^{\prime \prime}+2 u^{\prime}+7 u=0, \quad u(0)=2, \quad u^{\prime}(0)=1
$$
Find the smallest $T$ such that $|u(t)| \leq 0.1$ for all $t>T$. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In continuous time, the problem of finding a closed form solution for the state variable... | 71 | -273 | 4943.0 | 1.2 | 14.5115 | E |
Consider the initial value problem
$$
y^{\prime}=t y(4-y) / 3, \quad y(0)=y_0
$$
Suppose that $y_0=0.5$. Find the time $T$ at which the solution first reaches the value 3.98. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)... | 15 | 16 | 260.0 | -3.141592 | 3.29527 | E |
25. Consider the initial value problem
$$
2 y^{\prime \prime}+3 y^{\prime}-2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-\beta,
$$
where $\beta>0$.
Find the smallest value of $\beta$ for which the solution has no minimum point. | Although the first derivative (3x2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.) \sqrt[x]{x} Unique global maximum at x = e. (See figure at top of page.) x3 \+ 3x2 − 2x + 1 defined over the closed interval (segment) [−4,2] Local maximum at x = −1−/3, local minimum at x = −1+/3, globa... | 22 | 8.99 | 0.318 | 38 | 2 | E |
Consider the initial value problem
$$
y^{\prime}=t y(4-y) /(1+t), \quad y(0)=y_0>0 .
$$
If $y_0=2$, find the time $T$ at which the solution first reaches the value 3.99. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)... | 4.8 | 17 | 2.84367 | 131 | -32 | C |
28. A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
If the mass is to attain a velocity of no more than $10 \mathrm{~m} / \mathrm{s}$, find the maximum height from which it can be dropped. | If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere,... | 144 | 13.45 | 0.0182 | 0.6321205588 | 35 | B |
A home buyer can afford to spend no more than $\$ 800$ /month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
Determine the maximum amount that this buyer can affor... | Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be... | 1.61 | 21 | 0.36 | 89,034.79 | 1.25 | D |
A spring is stretched 6 in by a mass that weighs $8 \mathrm{lb}$. The mass is attached to a dashpot mechanism that has a damping constant of $0.25 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}$ and is acted on by an external force of $4 \cos 2 t \mathrm{lb}$.
If the given mass is replaced by a mass $m$, determine the valu... | As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in ... | 0.00539 | 420 | 4.0 | 35.2 | -0.041 | C |
A recent college graduate borrows $\$ 100,000$ at an interest rate of $9 \%$ to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $800(1+t / 120)$, where $t$ is the number of months since the loan was made.
Assuming that this payment schedule can be ma... | If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Cal... | -59.24 | 135.36 | 3.52 | 0.85 | -0.38 | B |
Consider the initial value problem
$$
y^{\prime}+\frac{1}{4} y=3+2 \cos 2 t, \quad y(0)=0
$$
Determine the value of $t$ for which the solution first intersects the line $y=12$. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found t... | 0.396 | 8.8 | 650000.0 | 10.065778 | 0.166666666 | D |
An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \$25 per month. Find the balance in the account after 3 years. | Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. Over one month, :\frac{0.1299 \tim... | 2283.63 | 0.082 | 1.91 | 6.1 | 35.2 | A |
A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
If the mass is dropped from a height of $30 \mathrm{~m}$, find its velocity when it hits the ground. | If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere,... | 0.75 | 11.58 | 1068.0 | 0.0408 | 3857 | B |
A mass of $100 \mathrm{~g}$ stretches a spring $5 \mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \mathrm{~cm} / \mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position. | The effective mass of the spring can be determined by finding its kinetic energy. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{... | 56 | 0.2244 | 6.07 | 12 | 0.66666666666 | B |
Suppose that a tank containing a certain liquid has an outlet near the bottom. Let $h(t)$ be the height of the liquid surface above the outlet at time $t$. Torricelli's principle states that the outflow velocity $v$ at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height $h$.
... | The results confirm the correctness of Torricelli's law very well. ==Discharge and time to empty a cylindrical vessel== Assuming that a vessel is cylindrical with fixed cross-sectional area A, with orifice of area A_A at the bottom, then rate of change of water level height dh/dt is not constant. From Torricelli's law,... | +4.1 | 0.16 | 130.41 | 3.51 | 1.5 | C |
Solve the initial value problem $y^{\prime \prime}-y^{\prime}-2 y=0, y(0)=\alpha, y^{\prime}(0)=2$. Then find $\alpha$ so that the solution approaches zero as $t \rightarrow \infty$. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Indeed, r... | −2 | -242.6 | 0.23333333333 | 20.2 | 6 | A |
If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ and if $W\left(y_1, y_2\right)(2)=3$, find the value of $W\left(y_1, y_2\right)(4)$. | Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. The Third Solution (, also known as Russicum) is a 1988 Italian crime-thriller film written and directed by Pasquale Squitieri and starring Treat Williams.VV.AA. Variety Film Reviews, Volume 18. In mathematical physics, the... | 0.3359 | 0.2307692308 | 1.602 | 0.65625 | 4.946 | E |
Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter. | University Library of Tromso, Ravnetrykk No. 29. , pp. 195–202. by exposing natural radium-226 to neutrons to produce radium-227, which decays with a 42-minute half-life to actinium-227. Actinium-227 (half-life 21.8 years) in turn decays via thorium-227 (half-life 18.7 days) to radium-223. Radium-223 (223Ra, Ra-223) is... | 479 | 672.4 | 2500.0 | 67 | -3.5 | B |
A tank originally contains $100 \mathrm{gal}$ of fresh water. Then water containing $\frac{1}{2} \mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \mathrm{~min}$ the process is stopped, and fresh water is... | The solution is 9 grams of sodium chloride (NaCl) dissolved in water, to a total volume of 1000 ml (weight per unit volume). The state (1, 2), for example, is impossible to reach from an initial state of (0, 0), since (1, 2) has both jugs partially full, and no reversible action is possible from this state. === Jug wit... | -0.38 | 0.2553 | 7.42 | 0.2553 | 1.3 | C |
A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously.
If $r=7.5 \%$, determine $k$ so that $\$ 1$ million will be available for retirement in 40 years. | The return over the five-year period for such an investor would be ($19.90 + $5.78) / $14.21 − 1 = 80.72%, and the arithmetic average rate of return would be 80.72%/5 = 16.14% per year. ==See also== * Annual percentage yield * Average for a discussion of annualization of returns * Capital budgeting * Compound annual gr... | 2.8108 | 3930 | 0.249 | 3.42 | +116.0 | B |
Consider the initial value problem
$$
y^{\prime \prime}+2 a y^{\prime}+\left(a^2+1\right) y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 .
$$
For $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value pro... | -75 | 24 | 2.0 | 7 | 1.8763 | E |
Consider the initial value problem
$$
y^{\prime \prime}+\gamma y^{\prime}+y=\delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0,
$$
where $\gamma$ is the damping coefficient (or resistance).
Find the time $t_1$ at which the solution attains its maximum value. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Springer-Verlag, Berlin, 2008. x+506 pp. * Protter, Murray H.; Weinberger, Hans F. Maximum principles in differential equations. A strong maximum principle for parabolic equations. Rearran... | -50 | 1.5377 | 0.0029 | 30 | 2.3613 | E |
Consider the initial value problem
$$
y^{\prime}+\frac{2}{3} y=1-\frac{1}{2} t, \quad y(0)=y_0 .
$$
Find the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} su... | 0.7812 | −1.642876 | 1.44 | -57.2 | -994.3 | B |
A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \mathrm{mg}$ of thorium-234 decays to $82.04 \mathrm{mg}$ in 1 week, determine the dec... | The decay constant is \frac{\ln(2)}{t_{1/2}} where "t_{1/2}" is the half-life of the radioactive material of interest. ==Example== The decay correct might be used this way: a group of 20 animals is injected with a compound of interest on a Monday at 10:00 a.m. (A simple way to check if you are using the decay correct f... | 6.6 | 0.1591549431 | 12.0 | 0.02828 | -9.54 | D |
Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\circ} \mathrm{F}$ when freshly ... | Rather, using today's terms, Newton noted after some mathematical manipulation that the rate of temperature change of a body is proportional to the difference in temperatures between the body and its surroundings. In that case, Newton's law only approximates the result when the temperature difference is relatively smal... | 2 | 9.90 | 0.375 | 6.07 | 16 | D |
Solve the initial value problem $4 y^{\prime \prime}-y=0, y(0)=2, y^{\prime}(0)=\beta$. Then find $\beta$ so that the solution approaches zero as $t \rightarrow \infty$. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Rearrange... | 0.321 | -1 | 76.0 | 7 | 11 | B |
Consider the initial value problem (see Example 5)
$$
y^{\prime \prime}+5 y^{\prime}+6 y=0, \quad y(0)=2, \quad y^{\prime}(0)=\beta
$$
where $\beta>0$.
Determine the smallest value of $\beta$ for which $y_m \geq 4$. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a giv... | 1.22 | 16.3923 | 14.0 | 71 | 0.444444444444444 | B |
A home buyer can afford to spend no more than $\$ 800 /$ month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
Determine the total interest paid during the term of ... | Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. When interest is continuously compounded, use \delta=n\ln{\left(1+\fr... | 102,965.21 | 29.9 | 0.84 | 4.0 | 25.6773 | A |
Find the fundamental period of the given function:
$$f(x)=\left\{\begin{array}{ll}(-1)^n, & 2 n-1 \leq x<2 n, \\ 1, & 2 n \leq x<2 n+1 ;\end{array} \quad n=0, \pm 1, \pm 2, \ldots\right.$$ | Period 2 is the first period in the periodic table from which periodic trends can be drawn. The constant function , where is independent of , is periodic with any period, but lacks a fundamental period. Since we are calculating a sine series, a_n=0\ \quad \forall n Now, b_n= \frac{2}{\pi} \int_0^\pi \cos(x)\sin(nx)\,\m... | 0.6321205588 | 4 | 7.136 | -194 | -0.10 | B |
A homebuyer wishes to finance the purchase with a \$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \$900? | thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. Mortgage calculators can be used to a... | 131 | 9.73 | 25.6773 | 135.36 | 0.0761 | B |
A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%? | Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be... | −2 | 8.87 | 15.757 | 27 | 804.62 | E |
Let a metallic rod $20 \mathrm{~cm}$ long be heated to a uniform temperature of $100^{\circ} \mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\circ} \mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. De... | The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. While thermal bars can form in both fall... | 2598960 | 7 | 35.91 | -9.54 | 310 | C |
Find $\gamma$ so that the solution of the initial value problem $x^2 y^{\prime \prime}-2 y=0, y(1)=1, y^{\prime}(1)=\gamma$ is bounded as $x \rightarrow 0$. | right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using ... | 2.8 | 2 | '-6.8' | 840 | 30 | B |
A tank contains 100 gal of water and $50 \mathrm{oz}$ of salt. Water containing a salt concentration of $\frac{1}{4}\left(1+\frac{1}{2} \sin t\right) \mathrm{oz} / \mathrm{gal}$ flows into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture in the tank flows out at the same rate.
The long-time behavi... | The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. If used correctly, the Leeson equation gives a useful prediction of oscillator performance in this range. In this case the phase difference is increasing, i... | 1590 | 0.24995 | '-1.46' | 10.4 | 9 | B |
A mass weighing $8 \mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\gamma$. Determine the value of $\gamma$ for which the system is critically damped; be sure to give the units for $\gamma$ | The graph shows the effect of a tuned mass damper on a simple spring–mass–damper system, excited by vibrations with an amplitude of one unit of force applied to the main mass, m1. Now considering m2 = , the blue line shows the motion of the damping mass and the red line shows the motion of the primary mass. By differen... | 8 | 1000 | '-1.78' | 773 | 0.366 | A |
Your swimming pool containing 60,000 gal of water has been contaminated by $5 \mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \mathrm{gal} / \mathrm{min}$... | 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. This depends on: the microorganism, the disinfectant being us... | -1368 | 7.136 | 3.0 | 9.8 | 0.36 | B |
For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity. If m = 10 kg, find the drag coefficient so tha... | The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficien... | -6.04697 | 0.5 | 273.0 | 0.0408 | 817.90 | D |
Consider the initial value problem
$$
3 u^{\prime \prime}-u^{\prime}+2 u=0, \quad u(0)=2, \quad u^{\prime}(0)=0
$$
For $t>0$ find the first time at which $|u(t)|=10$. | A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. These problems can come from a more typical initial value problem :u'(t) = f(u(t)), \qquad... | 10.7598 | 0.082 | 25.6773 | 1.5377 | 2 | A |
Consider the initial value problem
$$
9 y^{\prime \prime}+12 y^{\prime}+4 y=0, \quad y(0)=a>0, \quad y^{\prime}(0)=-1
$$
Find the critical value of $a$ that separates solutions that become negative from those that are always positive. | By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. More generally one might expect :f(\tau)=A \tau^k \left(1+b\tau ^{k_1} + \cdots\right) ==The most important critical exponents== Let us assume that the system has two different phases characterized by an order par... | 5.1 | 10 | 1.5 | 2688 | 0.0384 | C |
In an orienteering class, you have the goal of moving as far (straight-line distance) from base camp as possible by making three straight-line moves. You may use the following displacements in any order: (a) $\vec{a}, 2.0 \mathrm{~km}$ due east (directly toward the east); (b) $\vec{b}, 2.0 \mathrm{~km} 30^{\circ}$ nort... | The distance from the camp post to the market is (1/4-1/15), which will be repeated 3 times by the camel, and the maximum amount of bananas tranported to the market is 2-(1/4-1/15)*3=1.45 units; *# For m=1/2>(1/3+1/15), another midway camp post is necessary at a distance of 1/3 units from the first camp post, where a t... | 14.34457 | 13 | 4.8 | 140 | 0.18 | C |
"Top gun" pilots have long worried about taking a turn too tightly. As a pilot's body undergoes centripetal acceleration, with the head toward the center of curvature, the blood pressure in the brain decreases, leading to loss of brain function.
There are several warning signs. When the centripetal acceleration is $2 g... | Incidents of acceleration-induced loss of consciousness have caused fatal accidents in aircraft capable of sustaining high-g for considerable periods. Skilled pilots can use this loss of vision as their indicator that they are at maximum turn performance without losing consciousness. The danger of g-LOC to aircraft pil... | 0.132 | 1000 | 83.81 | 0.00017 | 5654.86677646 | C |
The world’s largest ball of string is about 2 m in radius. To the nearest order of magnitude, what is the total length L of the string in the ball? | If the string should be raised off the ground, all the way along the equator, how much longer would the string be? Even more surprising is that the size of the sphere or circle around which the string is spanned is irrelevant, and may be anything from the size of an atom to the Milky Way -- the result depends only on t... | -11.2 | 0.4207 | 2.0 | 479 | 0.245 | C |
You drive a beat-up pickup truck along a straight road for $8.4 \mathrm{~km}$ at $70 \mathrm{~km} / \mathrm{h}$, at which point the truck runs out of gasoline and stops. Over the next $30 \mathrm{~min}$, you walk another $2.0 \mathrm{~km}$ farther along the road to a gasoline station.
What is your overall displacement ... | A displacement may be also described as a relative position (resulting from the motion), that is, as the final position of a point relative to its initial position . Displacement is usually measured in units of tonnes or long tons. ==Definitions== There are terms for the displacement of a vessel under specified conditi... | 10.4 | 258.14 | 3.0 | 273 | -0.347 | A |
A heavy object can sink into the ground during an earthquake if the shaking causes the ground to undergo liquefaction, in which the soil grains experience little friction as they slide over one another. The ground is then effectively quicksand. The possibility of liquefaction in sandy ground can be predicted in terms o... | Eng., 139(3), 407–419. http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000743 ] #The earthquake load, measured as cyclic stress ratio CSR=\frac{\tau_{av}}{\sigma'_{v}}=0,65\frac{a_{max}}{g}\frac{\sigma_{v}}{\sigma'_{v}}r_d Evaluation of soil liquefaction from surface analysis #the capacity of the soil to resist liquefact... | 0.70710678 | 2598960 | '-233.0' | 0.9966 | 1.4 | E |
What is the angle $\phi$ between $\vec{a}=3.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}$ and $\vec{b}=$ $-2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{k}}$ ? (Caution: Although many of the following steps can be bypassed with a vector-capable calculator, you will learn more about scalar products if, at least here, you use these s... | ===Vector rejection=== By definition, \mathbf{a}_2 = \mathbf{a} - \mathbf{a}_1 Hence, \mathbf{a}_2 = \mathbf{a} - \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}}. ==Properties== ===Scalar projection=== The scalar projection on is a scalar which has a negative sign if 90 degrees < θ ≤ 180 d... | 0.03 | 804.62 | 109.0 | 2.7 | 8 | C |
During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement $\vec{d}=(-3.0 \mathrm{~m}) \hat{\mathrm{i}}$ while a steady wind pushes against the crate with a force $\vec{F}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}+(-6.0 \mathrm{~N}) \hat{\mathrm{j}}$. The situation and coordinate axes ... | Kinetic energy is the movement energy of an object. The velocity v of the car can be determined from the length of the skid using the work–energy principle, kWs = \frac{W}{2g} v^2,\quad\text{or}\quad v = \sqrt{2ksg}. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. The kine... | -383 | 11 | 2.0 | 4.0 | 1.602 | D |
When the force on an object depends on the position of the object, we cannot find the work done by it on the object by simply multiplying the force by the displacement. The reason is that there is no one value for the force-it changes. So, we must find the work in tiny little displacements and then add up all the work ... | The work of the net force is calculated as the product of its magnitude and the particle displacement. The small amount of work by the forces over the small displacements can be determined by approximating the displacement by so \delta W = \mathbf{F}_1\cdot\mathbf{V}_1\delta t+\mathbf{F}_2\cdot\mathbf{V}_2\delta t + \l... | 0.14 | 210 | 5.9 | 1855 | 7.0 | E |
The charges of an electron and a positron are $-e$ and $+e$. The mass of each is $9.11 \times 10^{-31} \mathrm{~kg}$. What is the ratio of the electrical force to the gravitational force between an electron and a positron?
| The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. While the electron has a negative electric charge, the positron has a positive electric charge, and is produced naturally in certain types of radioactive decay. It turns out that ... | 4.16 | 5.51 | 27.211 | -36.5 | 7.136 | A |
In Fig. 21-26, particle 1 of charge $+q$ and particle 2 of charge $+4.00 q$ are held at separation $L=9.00 \mathrm{~cm}$ on an $x$ axis. If particle 3 of charge $q_3$ is to be located such that the three particles remain in place when released, what must be the $x$ coordinate of particle 3?
| The position coordinates xj and xk are replaced > by their relative position rjk = xj − xk and by the vector to their center > of mass Rjk = (mj qj \+ mkqk)/(mj \+ mk). thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. A trion is a lo... | 0.9731 | 313 | 3.0 | 0.14 | 12 | C |
Two charged particles are fixed to an $x$ axis: Particle 1 of charge $q_1=2.1 \times 10^{-8} \mathrm{C}$ is at position $x=20 \mathrm{~cm}$ and particle 2 of charge $q_2=-4.00 q_1$ is at position $x=70 \mathrm{~cm}$. At what coordinate on the axis (other than at infinity) is the net electric field produced by the two p... | Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The particle located experiences an interaction with the electric field. Here and are the charges on particles 1 and 2 respectively, and are the masses ... | 200 | 5040 | 4.86 | -30 | 0.6321205588 | D |
The volume charge density of a solid nonconducting sphere of radius $R=5.60 \mathrm{~cm}$ varies with radial distance $r$ as given by $\rho=$ $\left(14.1 \mathrm{pC} / \mathrm{m}^3\right) r / R$. What is the sphere's total charge? | Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \f... | 0.332 | 170 | 0.6 | 7.78 | 292 | D |
Two charged concentric spher-
ical shells have radii $10.0 \mathrm{~cm}$ and $15.0 \mathrm{~cm}$. The charge on the inner shell is $4.00 \times 10^{-8} \mathrm{C}$, and that on the outer shell is $2.00 \times 10^{-8} \mathrm{C}$. Find the electric field at $r=12.0 \mathrm{~cm}$. | The "spherium" model consists of two electrons trapped on the surface of a sphere of radius R. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring bet... | 2.50 | 0.396 | 0.0182 | 0.00539 | 0.686 | A |
Assume that a honeybee is a sphere of diameter 1.000 $\mathrm{cm}$ with a charge of $+45.0 \mathrm{pC}$ uniformly spread over its surface. Assume also that a spherical pollen grain of diameter $40.0 \mu \mathrm{m}$ is electrically held on the surface of the bee because the bee's charge induces a charge of $-1.00 \mathr... | thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This approximation has been used to estimate the force between two charged colloidal particles ... | 4 | 1.8763 | '-1.49' | 2.6 | 7 | D |
In the radioactive decay of Eq. 21-13, $\mathrm{a}^{238} \mathrm{U}$ nucleus transforms to ${ }^{234} \mathrm{Th}$ and an ejected ${ }^4 \mathrm{He}$. (These are nuclei, not atoms, and thus electrons are not involved.) When the separation between ${ }^{234} \mathrm{Th}$ and ${ }^4 \mathrm{He}$ is $9.0 \times 10^{-15} \... | See also: H. Geiger and J.M. Nuttall (1912) "The ranges of α particles from uranium," Philosophical Magazine, Series 6, vol. 23, no. 135, pages 439-445. in its modern form the Geiger–Nuttall law is :\log_{10}T_{1/2}=\frac{A(Z)}{\sqrt{E}}+B(Z) where T_{1/2} is the half-life, E the total kinetic energy (of the alpha part... | -2 | 449 | 5.1 | 1.8 | 228 | C |
The electric field in an $x y$ plane produced by a positively charged particle is $7.2(4.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}$ at the point $(3.0,3.0) \mathrm{cm}$ and $100 \hat{\mathrm{i}} \mathrm{N} / \mathrm{C}$ at the point $(2.0,0) \mathrm{cm}$. What is the $x$ coordinate of the particl... | Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting... | 0.0625 | 35 | '-1.0' | 344 | 54.394 | C |
A charge distribution that is spherically symmetric but not uniform radially produces an electric field of magnitude $E=K r^4$, directed radially outward from the center of the sphere. Here $r$ is the radial distance from that center, and $K$ is a constant. What is the volume density $\rho$ of the charge distribution? | Let the first charge distribution \rho_1(\mathbf{r}') be centered on the origin and lie entirely within the second charge distribution \rho_2(\mathbf{r}'). In the interior case, where r' > r, the result is: \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^\ell I_{\ell m} r^\ell \sqrt{\... | 0.71 | 6 | 0.36 | 52 | 157.875 | B |
Two particles, each with a charge of magnitude $12 \mathrm{nC}$, are at two of the vertices of an equilateral triangle with edge length $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the third vertex if both charges are positive? | thumb|upright=1.25|triangle ABC exsymmedians (red): e_a, e_b, e_c symmedians (green): s_a, s_b, s_c exsymmedian points (red): E_a, E_b, E_c The exsymmedians are three lines associated with a triangle. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{... | 0.0000092 | 9 | 0.8185 | 47 | -11.875 | D |
How much work is required to turn an electric dipole $180^{\circ}$ in a uniform electric field of magnitude $E=46.0 \mathrm{~N} / \mathrm{C}$ if the dipole moment has a magnitude of $p=3.02 \times$ $10^{-25} \mathrm{C} \cdot \mathrm{m}$ and the initial angle is $64^{\circ} ?$
| This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. For a spatially uniform electric field across the small region occupied by the dipole, the energy U and the torque \boldsymbol{\tau} are given by U = - \mathbf{... | 1.16 | 1.22 | 0.7812 | 3.54 | -32 | B |
We know that the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that these magnitudes differ from each other by $0.00010 \%$. With what force would two copper coins, placed $1.0 \mathrm{~m}$ apart, repel each other? Assume that each coin contains $3 \times 10^{22}$ co... | This is not the case with copper. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The range of these forces is typically below 1 nm. ==Like-cha... | 2600 | 1.7 | 12.0 | 71 | 21 | B |
What must be the distance between point charge $q_1=$ $26.0 \mu \mathrm{C}$ and point charge $q_2=-47.0 \mu \mathrm{C}$ for the electrostatic force between them to have a magnitude of $5.70 \mathrm{~N}$ ?
| The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r i... | -45 | 7.27 | 0.000226 | 1.61 | 1.39 | E |
Three charged particles form a triangle: particle 1 with charge $Q_1=80.0 \mathrm{nC}$ is at $x y$ coordinates $(0,3.00 \mathrm{~mm})$, particle 2 with charge $Q_2$ is at $(0,-3.00 \mathrm{~mm})$, and particle 3 with charge $q=18.0$ $\mathrm{nC}$ is at $(4.00 \mathrm{~mm}, 0)$. In unit-vector notation, what is the elec... | The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r i... | 0.829 | 11 | '-0.5' | -214 | 0.14 | A |
A particle of charge $-q_1$ is at the origin of an $x$ axis. (a) At what location on the axis should a particle of charge $-4 q_1$ be placed so that the net electric field is zero at $x=2.0 \mathrm{~mm}$ on the axis? | The figure at right shows the electric field lines of two equal charges of opposite sign. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The net electrostatic force acting on a charged particle with ... | 3.42 | 6.0 | '-114.4' | 11 | 4 | B |
An electron is released from rest in a uniform electric field of magnitude $2.00 \times 10^4 \mathrm{~N} / \mathrm{C}$. Calculate the acceleration of the electron. (Ignore gravitation.) | Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. For the special ... | 16 | -3.141592 | '-0.029' | 3.51 | 0.2 | D |
Identify $\mathrm{X}$ in the following nuclear reactions: (a) ${ }^1 \mathrm{H}+$ ${ }^9 \mathrm{Be} \rightarrow \mathrm{X}+\mathrm{n} ;$ (b) ${ }^{12} \mathrm{C}+{ }^1 \mathrm{H} \rightarrow \mathrm{X} ;$ (c) ${ }^{15} \mathrm{~N}+{ }^1 \mathrm{H} \rightarrow{ }^4 \mathrm{He}+\mathrm{X}$. Appendix F will help. | Jonathan Feng et al. attribute this 6.8-σ anomaly to a 17 MeV protophobic X-boson dubbed the X17 particle. The X17 particle is a hypothetical subatomic particle proposed by Attila Krasznahorkay and his colleagues to explain certain anomalous measurement results. Krasznahorkay (2019) posted a preprint announcing that he... | 0.366 | 0.020 | null | -87.8 | 48 | C |
The nucleus of a plutonium-239 atom contains 94 protons. Assume that the nucleus is a sphere with radius $6.64 \mathrm{fm}$ and with the charge of the protons uniformly spread through the sphere. At the surface of the nucleus, what are the magnitude of the electric field produced by the protons? | This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... The proton radius is approximately one femto... | 54.394 | 6.3 | 25.6773 | 1000 | 3.07 | E |
A nonconducting spherical shell, with an inner radius of $4.0 \mathrm{~cm}$ and an outer radius of $6.0 \mathrm{~cm}$, has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density $\rho$ is the charge per unit volume, with the unit coulomb per cubic meter. For this s... | It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\... | 0.22222222 | 0.32 | 1838.50666349 | +37 | 3.8 | E |
An electron is shot directly
Figure 23-50 Problem 40. toward the center of a large metal plate that has surface charge density $-2.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^2$. If the initial kinetic energy of the electron is $1.60 \times 10^{-17} \mathrm{~J}$ and if the electron is to stop (due to electrostatic repulsi... | thumb|upright=1.3|right|Launch of Electron in start of the "Birds of the Feather" mission. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. Electron is a two-stage small-lift launch vehicle built and operated by Rocket Lab. The objective of the Thomson problem is to determine the ... | 1.5 | 0.42 | 0.44 | -0.5 | 4152 | C |
A square metal plate of edge length $8.0 \mathrm{~cm}$ and negligible thickness has a total charge of $6.0 \times 10^{-6} \mathrm{C}$. Estimate the magnitude $E$ of the electric field just off the center of the plate (at, say, a distance of $0.50 \mathrm{~mm}$ from the center) by assuming that the charge is spread unif... | We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian ... | 10.8 | 0.70710678 | 2.3613 | -0.16 | 5.4 | E |
A neutron consists of one "up" quark of charge $+2 e / 3$ and two "down" quarks each having charge $-e / 3$. If we assume that the down quarks are $2.6 \times 10^{-15} \mathrm{~m}$ apart inside the neutron, what is the magnitude of the electrostatic force between them?
| Neutrons and protons, both nucleons, are affected by the nuclear force almost identically. This same force is much weaker between neutrons and protons, because it is mostly neutralized within them, in the same way that electromagnetic forces between neutral atoms (van der Waals forces) are much weaker than the electrom... | 0 | 47 | 635013559600.0 | 0.5768 | 3.8 | E |
In an early model of the hydrogen atom (the Bohr model), the electron orbits the proton in uniformly circular motion. The radius of the circle is restricted (quantized) to certain values given by where $a_0=52.92 \mathrm{pm}$. What is the speed of the electron if it orbits in the smallest allowed orbit? | In the simplest atom, hydrogen, a single electron orbits the nucleus, and its smallest possible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. This means that the innermost electrons orbit at approximately 1/2 the Bohr radius. Since the reduced mass of the electron–proton system i... | 7.0 | 3.23 | 4.0 | 2.19 | 0.064 | D |
At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius $0.600 \mathrm{~m}$ is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk? | thumb|upright=1.35|An ellipse, its minimum bounding box, and its director circle. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. An obvious choice for this fraction is ½ (−3 dB), in which case the diameter obtaine... | 15.757 | 0.346 | 0.166666666 | 420 | 0 | B |
Of the charge $Q$ on a tiny sphere, a fraction $\alpha$ is to be transferred to a second, nearby sphere. The spheres can be treated as particles. What value of $\alpha$ maximizes the magnitude $F$ of the electrostatic force between the two spheres? | The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Related problems include the study of ... | 12 | 3857 | '-4564.7' | 0.5 | 6760000 | D |
In a spherical metal shell of radius $R$, an electron is shot from the center directly toward a tiny hole in the shell, through which it escapes. The shell is negatively charged with a surface charge density (charge per unit area) of $6.90 \times 10^{-13} \mathrm{C} / \mathrm{m}^2$. What is the magnitude of the electro... | The radius r is then defined to be the classical electron radius, r_\text{e}, and one arrives at the expression given above. This asymmetric distribution of charge within the particle gives rise to a small negative squared charge radius for the particle as a whole. The electrostatic potential at a distance r from a cha... | 0 | 49 | 2.0 | 96.4365076099 | 243 | A |
A particle of charge $+3.00 \times 10^{-6} \mathrm{C}$ is $12.0 \mathrm{~cm}$ distant from a second particle of charge $-1.50 \times 10^{-6} \mathrm{C}$. Calculate the magnitude of the electrostatic force between the particles. | If is the distance between the charges, the magnitude of the force is |\mathbf{F}|=\frac{|q_1q_2|}{4\pi\varepsilon_0 r^2}, where is the electric constant. The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the pr... | 1.39 | 0.1591549431 | 3.8 | 5.7 | 2.81 | E |
A charged particle produces an electric field with a magnitude of $2.0 \mathrm{~N} / \mathrm{C}$ at a point that is $50 \mathrm{~cm}$ away from the particle. What is the magnitude of the particle's charge?
| Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The voltage between two points is defined as:Lerner L, Physics for scientists and engineers, Jones & Bartlett, 1997, pp. 685–686 {\Delta V} = -\int {\ma... | 8 | 1.16 | 56.0 | 30 | 2.81 | C |
In Millikan's experiment, an oil drop of radius $1.64 \mu \mathrm{m}$ and density $0.851 \mathrm{~g} / \mathrm{cm}^3$ is suspended in chamber C (Fig. 22-16) when a downward electric field of $1.92 \times 10^5 \mathrm{~N} / \mathrm{C}$ is applied. Find the charge on the drop, in terms of $e$. | Using the known electric field, Millikan and Fletcher could determine the charge on the oil droplet. right|thumb|Millikan's setup for the oil drop experiment|300x300px The oil drop experiment was performed by Robert A. Millikan and Harvey Fletcher in 1909 to measure the elementary electric charge (the charge of the ele... | 2.00 | 12 | 0.0 | 62.8318530718 | -5 | E |
The charges and coordinates of two charged particles held fixed in an $x y$ plane are $q_1=+3.0 \mu \mathrm{C}, x_1=3.5 \mathrm{~cm}, y_1=0.50 \mathrm{~cm}$, and $q_2=-4.0 \mu \mathrm{C}, x_2=-2.0 \mathrm{~cm}, y_2=1.5 \mathrm{~cm}$. Find the magnitude of the electrostatic force on particle 2 due to particle 1. | The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r i... | 35 | -3.141592 | '-1.0' | 2 | 0.15 | A |
An electron on the axis of an electric dipole is $25 \mathrm{~nm}$ from the center of the dipole. What is the magnitude of the electrostatic force on the electron if the dipole moment is $3.6 \times 10^{-29} \mathrm{C} \cdot \mathrm{m}$ ? Assume that $25 \mathrm{~nm}$ is much larger than the separation of the charged p... | The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the ... | 6.6 | 0.5 | 0.241 | 1.8 | 30 | A |
Earth's atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total s... | Consequently, there is always a small current of approximately 2pA per square metre transporting charged particles in the form of atmospheric ions between the ionosphere and the surface. === Fair weather=== This current is carried by ions present in the atmosphere (generated mainly by cosmic rays in the free tropospher... | 0.318 | 11.58 | 3.03 | 122 | 460.5 | D |
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