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 |
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