1.The work done by a force along a path is written most generally as

1. $Fd$
2. $Fd\cos\theta$
3. $\int \vec{F}\cdot d\vec{r}$
4. $\tfrac12 mv^2$

2.For a conservative force in one dimension, the force is recovered from the potential energy by

1. $F=U/x$
2. $F=-\dfrac{dU}{dx}$
3. $F=\dfrac{dU}{dx}$
4. $F=\displaystyle\int U\,dx$

3.Instantaneous power is written as

1. $W t$
2. $\dfrac{dE}{dt}$
3. $\dfrac{dt}{dE}$
4. $\vec{F}\cdot d\vec{r}$

4.On a graph of $U(x)$, a point where the curve has a minimum is

1. an unstable equilibrium
2. a stable equilibrium
3. a turning point
4. a point where the force is largest

5.A particle has total energy $E$ and moves in a potential $U(x)$. A turning point is a place where

1. $U=0$
2. $U=E$
3. $dU/dx=0$
4. $d^2U/dx^2=0$

6.A force $\vec{F}$ acts on a particle whose displacement is $d\vec{r}$. The dot product $\vec{F}\cdot d\vec{r}$ is zero when

1. the force is zero only
2. the displacement is zero only
3. the force and the displacement are at a right angle, or either is zero
4. the force is constant

7.Which statement identifies a conservative force?

1. It always does positive work
2. The work it does around any closed path is zero
3. It never depends on position
4. It acts only between solid surfaces

8.The potential energy of a conservative force is defined only up to

1. a multiplying factor
2. an added constant
3. a sign
4. a factor of the mass

9.For a force acting on a particle moving with velocity $\vec{v}$, the instantaneous power is

1. $\vec{F}\times\vec{v}$
2. $\vec{F}\cdot\vec{v}$
3. $\dfrac{F}{v}$
4. $\tfrac12 Fv^2$

10.A particle moves once around a closed loop under a single force and returns to its starting point with a different speed. What follows?

1. The force is conservative
2. The force is not conservative
3. The force did no work
4. The kinetic energy is not defined

11.At a point on a $U(x)$ curve where the slope is large and positive, the force on the particle is

1. large and in the $+x$ direction
2. large and in the $-x$ direction
3. zero
4. small and in the $+x$ direction

12.A particle sits exactly at a maximum of $U(x)$. Which description is correct?

1. It is in stable equilibrium, because the force is zero
2. It is in unstable equilibrium, because the force is zero but any nudge drives it away
3. It is at a turning point, because the kinetic energy is zero
4. It cannot be in equilibrium, because $U$ is not zero

13.Starting from $\vec{F}=m\,d\vec{v}/dt$, the work-energy theorem is obtained by

1. integrating the force with respect to time
2. integrating the force with respect to displacement and changing the variable to $v$
3. differentiating the kinetic energy with respect to position
4. assuming the force is constant

14.Two potential energy functions differ by a constant. Compared with the first, the second gives

1. a different force and different turning points
2. the same force but different turning points for a given total energy $E$
3. a different force but the same turning points
4. the same force and the same motion, once $E$ is shifted by the same constant

15.An engine delivers constant power to a car on a level road with no resistance. As the speed rises, the forward force

1. rises
2. stays the same
3. falls
4. reverses

16.A force $F(x)=6x$ N acts along $x$. How much work does it do from $x=0$ to $x=2.0$ m?

1. $6.0$ J
2. $12$ J
3. $18$ J
4. $24$ J

17.A particle has $U(x)=4x^2$ J with $x$ in meters. What is the force at $x=3.0$ m?

1. $-24$ N
2. $-12$ N
3. $+12$ N
4. $+24$ N

18.A force of $20$ N acts on a particle moving at $3.0$ m/s in the same direction. What is the instantaneous power?

1. $6.7$ W
2. $23$ W
3. $60$ W
4. $180$ W

19.A force $F(x)=3x^2$ N acts along $x$. How much work does it do from $x=1.0$ m to $x=3.0$ m?

1. $9.0$ J
2. $18$ J
3. $26$ J
4. $27$ J

20.A constant force $\vec{F}=(4\hat{\imath}+3\hat{\jmath})$ N acts while a particle moves through $\Delta\vec{r}=(2\hat{\imath}-1\hat{\jmath})$ m. How much work is done?

1. $5.0$ J
2. $8.0$ J
3. $11$ J
4. $14$ J

21.A conservative force is $F(x)=-kx$. Taking $U=0$ at $x=0$, the potential energy is

1. $-kx$
2. $-\tfrac12 kx^2$
3. $\tfrac12 kx^2$
4. $kx^2$

22.The energy delivered to a system is $E(t)=5t^3$ J with $t$ in seconds. What is the instantaneous power at $t=2.0$ s?

1. $20$ W
2. $40$ W
3. $60$ W
4. $120$ W

23.A particle moves in $U(x)=2x^2-16x$ J. Where is the equilibrium point?

1. $x=0$
2. $x=2.0$ m
3. $x=4.0$ m
4. $x=8.0$ m

24.A $2.0$ kg particle speeds up from $3.0$ m/s to $5.0$ m/s. What is the net work done on it?

1. $4.0$ J
2. $8.0$ J
3. $16$ J
4. $32$ J

25.A particle has $U(x)=\dfrac{A}{x}$ with $A$ a positive constant. The force on the particle is

1. $-\dfrac{A}{x^2}$
2. $+\dfrac{A}{x^2}$
3. $-\dfrac{A}{x}$
4. $+A\ln x$

26.A force $F(x)=8-2x$ N acts on a particle moving along $x$. How much work does it do from $x=0$ to $x=6.0$ m?

1. $0$ J
2. $12$ J
3. $24$ J
4. $48$ J

27.A $1.0$ kg particle moves in $U(x)=x^3-6x^2+9x$ J with total energy $E=4.0$ J. Which value of $x$ is a turning point?

1. $x=1.0$ m
2. $x=2.0$ m
3. $x=3.0$ m
4. $x=4.0$ m

28.A $3.0$ kg block starts from rest at $x=0$ and a force $F(x)=12x$ N pushes it along a frictionless surface. How fast is it moving at $x=2.0$ m?

1. $2.8$ m/s
2. $4.0$ m/s
3. $5.7$ m/s
4. $8.0$ m/s

29.A particle moves with $\vec{v}=(3\hat{\imath}+4\hat{\jmath})$ m/s while a force $\vec{F}=(2\hat{\imath}-1\hat{\jmath})$ N acts on it. What is the instantaneous power?

1. $-2.0$ W
2. $2.0$ W
3. $10$ W
4. $25$ W

30.A particle moves under $F(x)=-\dfrac{6}{x^3}$ N. Taking $U\to 0$ as $x\to\infty$, the potential energy is

1. $-\dfrac{3}{x^2}$
2. $+\dfrac{3}{x^2}$
3. $-\dfrac{6}{x^2}$
4. $+\dfrac{2}{x^2}$

1.State the general definition of the work done by a force along a path, and say what has to be true before it collapses to $Fd\cos\theta$.

2.Explain in words what the minus sign in $F=-dU/dx$ is doing.

3.Give the two expressions for instantaneous power used in this unit, and say when each is the convenient one.

4.On a graph of $U(x)$, say how to find every equilibrium point by eye.

5.Explain how the shape of $U(x)$ near a flat point decides whether the equilibrium is stable or unstable.

6.Show in words why a force that is always perpendicular to the velocity can never change the speed of a particle.

7.Describe how to get $U(x)$ from a conservative $F(x)$, and say what the limits of the integral mean.

8.State the closed path test for a conservative force and explain why it is equivalent to path independence.

9.A car engine delivers constant power. Explain why the acceleration falls off as the car speeds up.

10.A particle has total energy $E$ in a potential $U(x)$. Explain why the motion is confined to the regions where $U\le E$.

11.Outline the derivation of the work-energy theorem from Newton’s second law.

12.Explain why a potential energy can be written for gravity but not for kinetic friction.

13.Two students choose different places for $U=0$ in the same problem. Explain what differs in their answers and what does not.

14.Explain what the area under a graph of $F$ against $x$ represents, and what a region below the axis contributes.

15.A potential energy curve has a minimum, then a maximum, then falls away forever. Describe the possible motions of a particle for a total energy below the maximum and for a total energy above it.

16.Starting from $U(x)=\tfrac12 kx^2$, obtain the force and explain why the result is consistent with what the spring does physically.

17.A particle travels a closed loop under two forces, one conservative and one not. State the net work done by each around the loop and what that implies for the kinetic energy at the end.

18.Power can be written as $dE/dt$ and as $\vec{F}\cdot\vec{v}$. Show that the two are the same statement.

19.For the gravitational force between two masses, explain why the form $U=-GMm/r$ and the form $U=mgh$ do not contradict each other.

20.Explain how to read the size and the direction of the force at any point directly off a graph of $U(x)$, and say what a steep downhill stretch means for the particle.