1. Work is the ____________ transferred to or from an object by a force acting on the object.

2. True or False An interpretation of the line integral $\int_{C}\mathbf{F\,{\cdot}\,}d\mathbf{r}$ is the work done by the force $\mathbf{F}$ in moving an object along a piecewise-smooth curve $C$.

3. True or False The work done by a force $\mathbf{F}$ in moving an object from $A$ to $B$ equals the change in the potential energy of the object from $A$ to $B$.

4. Multiple Choice In a conservative force field $\mathbf{F}$, the work $W$ done by $\mathbf{F}$ in moving an object along a closed, piecewise smooth curve $C$ is [(a) a maximum, (b) a minimum, (c) $\left\Vert \mathbf{F}\right\Vert ,$ (d) $0$].

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5. Kinetic energy $K$ is the energy of ____________; potential energy $U$ is the energy of ____________.

6. True or False In a conservative force field $\mathbf{F}$, the difference of the potential energy $U$ and the kinetic energy $K$ equals $0$.

7. $\mathbf{F}=y\mathbf{i}+x\mathbf{j}$ along $\mathbf{r}(t)=t \mathbf{i}+t^{2}\mathbf{j}$ from $t=0$ to $t=1$

8. $\mathbf{F}=xy\mathbf{i}+y^{2}\mathbf{j}$ along $\mathbf{r} (t)=t\mathbf{i}+t^{2}\mathbf{j}$ from $t=0$ to $t=1$

9. $\mathbf{F}=(x-2y)\mathbf{i}+xy\mathbf{j}$ along $\mathbf{r} (t)=3\cos t\,\mathbf{i}+2\sin t\,\mathbf{j}$ from $t=0$ to $t=\dfrac{\pi }{2}$

10. $\mathbf{F}=x\mathbf{i}-y\mathbf{j}$ along $\mathbf{r} (t)=\cos t\,\mathbf{i}+\sin t\,\mathbf{j}$ from $t=0$ to $t=2\pi$

11. $\mathbf{F}=x^{2}y\mathbf{i}+(x^{2}-y^{2})\mathbf{j}$ along $y=2x^{2}$ from $(0,0)$ to $(1,2)$

12. $\mathbf{F}=y\sin x\,\mathbf{i}-x\cos y\,\mathbf{j}$ along $y=x$ from $(0,0)$ to $(1,1)$

13. $\mathbf{F}=(y-x^{2})\mathbf{i}+x\mathbf{j}$ along the upper half of the circle $x^{2}+y^{2}=1$ from $(1,0)$ to $(-1,0)$

14. $\mathbf{F}=(y-x^{2})\mathbf{i}+x\mathbf{j}$ along the line segments joining $(1,0)$ to $(1,1)$ to $(-1,1)$ to $(-1,0)$

15. $\mathbf{F}=y^{3}\mathbf{i}+x^{3}\mathbf{j}$ along the ellipse $\mathbf{r}(t)=a\cos t\,\mathbf{i}+b\sin t\,\mathbf{j}$ from $t=0$ to $t=2\pi$

16. $\mathbf{F}=-y^{2}\mathbf{i}+x^{2}\mathbf{j}$ along the upper half of the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ from $(a,0)$ to $(-a,0)$

17. $\mathbf{F}=3x\mathbf{i}+2y\mathbf{j}$

    

18. $\mathbf{F}=2xy\mathbf{i}+xy^{2}\mathbf{j}$

    

19. $\mathbf{F}=e^{x}y\mathbf{i}+e^{x}\mathbf{j}$

    

20. $\mathbf{F}=e^{x}\mathbf{i}+5y\mathbf{j}$

    

21. Verify that the force field $\mathbf{F}=y\mathbf{i} -x\mathbf{j}$ is not a conservative vector field. Show that the integral $\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}$ is dependent on the path of integration by taking two paths in which the starting point is the origin $(0,0)$ and the endpoint is $(1,1)$. For one path, take the line $x=y$. For the other path, take the $x$-axis out to the point $(1,0)$ and then the line $x=1$ up to the point $(1,1)$.

22. Work Find the work done by a variable force $\mathbf{F} (x,y) =( x^{2}\mathbf{i}+y\mathbf{j})$ to move an object along $C$ traced out by the parametric equations $x=t$, $y=t^{2},$ $0\leq t\leq 3.$

23. Work Find the work done by a variable force $\mathbf{F} (x,y) =\left( \cos x\mathbf{i}+\sin y\mathbf{j}\right)$ to move an object along the curve $C$ traced out by the parametric equations $x=t$, $y=\sqrt{t},$ $0\leq t\leq \pi$.

24. Work An object moves in a clockwise direction from the origin to the point $(2a,0)$ along the upper half of the circle $(x-a)^{2}+y^{2}=a^{2}$ and is acted upon by a force $\mathbf{F}$ with constant magnitude $3$ and with direction $\mathbf{i}+\mathbf{j}$. Find the work $W$ done by this force.

25. Work An object moves along the curve $\mathbf{r}(t)=64\sqrt{3}t\,\mathbf{i}+(64t-16t^{2})\mathbf{j}$ from $t=0$ to $t=4$ and is acted upon by a force $\mathbf{F}$ whose magnitude is directly proportional to the speed of the object and whose direction is opposite to that of the velocity. Find the work $W$ done by this force.

26. Minimizing Work An object moves from the point $(0,0)$ to the point $(1,0)$ along the curve $y=ax(1-x)$ and is acted upon by a force given by $\mathbf{F}(x,y)=(y^{2}+1)\mathbf{i}+(x+y)\mathbf{j}$. Find $a$ so that the work $W$ done is a minimum.

27. Molecular Work The repelling force between a charged particle $P$ at the origin and an oppositely charged particle $Q$ at $(x,y)$ is $$\mathbf{F}(x,y)=\dfrac{x}{(x^{2}+y^{2})^{3/2}}\mathbf{i}+\dfrac{y}{ (x^{2}+y^{2})^{3/2}}\mathbf{j}$$

    Find the work done by $\mathbf{F}$ as $Q$ moves along the line segment from $(1,0)$ to $(-1,2)$.

28. Work Find the work $W$ done by the force field $\mathbf{F} ( x,y,z) =x\mathbf{i}+y\mathbf{j}+\mathbf{k}$ in moving an object along the helix $\mathbf{r}=\mathbf{r}(t) =\sin t\,\mathbf{i}+\cos t\mathbf{j}+t\mathbf{k}$, $0\leq t\leq \pi$.

29. Work Find the work $W$ done by the force field $\mathbf{F} ( x,y,z) =x^{2}\mathbf{i}+y^{2}\mathbf{j}+z^{2}\mathbf{k}$ in moving an object along the helix $\mathbf{r}=\mathbf{r}(t)=2\cos t\,\mathbf{i}+2\sin t\mathbf{j}+2t\mathbf{k}$, $0\leq t\leq \pi$.

30. Work Suppose sea level is the $xy$-plane and the $z$-axis points upward. Then the gravitational force on an object of mass $m$ near sea level may be taken to be $\mathbf{F}=mg\mathbf{k}$, where $g$ is the acceleration due to gravity. Show that the work done by gravity on an object moving from $(x_{1},y_{1},z_{1})$ to $(x_{2},y_{2},z_{2})$ along any path is $W=mg(z_{2}-z_{1})$.

31. Work

    1. (a) Suppose an object travels through a force field along a path that is always normal to the force. Show that the work done in moving the object from one point to another is $0$.
    2. (b) Why does it follow that the gravitational field of the earth does no work on a satellite in circular orbit?

32. Work Show that if the work done by a force $\mathbf{F}(x,y)$ in moving an object along any piecewise-smooth, closed curve in an open set $S$ in the plane is $0$, then the work done by $\mathbf{F}$ in moving an object from a point $A$ to a point $B$ in $S$ is independent of the piecewise-smooth path chosen.

33. Escape Speed

    1. (a)

       Show that the minimum speed needed to escape from the surface of a planet of mass $M$ in kilograms and radius $R$ in meters is $v_{esc}=\sqrt{\dfrac{2GM}{R}}$, where $G=6.67\times 10^{-11} {\rm N} {\rm m}^{2}/ {\rm kg}^{2}$.

       This speed is called the escape speed for the planet.

       (Hint: Use the relationship between kinetic energy $K$ and the work $W$ done by a force $\mathbf{F}$.)

    2. (b) Find the escape speed for Earth (in kilometers per second) using $M_{\rm Earth}=5.97\times 10^{24} {\rm kg}$ and $R_{\rm Earth}=6.38\times 10^{6} {\rm m}$.
    3. (c) Find the escape speed for Mars (in kilometers per second) using $M_{\rm Mars}=6.42\times 10^{23} {\rm kg}$ and $R_{\rm Mars}=3.40\times 10^{6}{\rm m}$.
    4. (d) Find the escape speed for the Sun (in kilometers per second) using $M_{\rm Sun}=1.99\times 10^{30} {\rm kg}$ and $R_{\rm Sun}=6.96\times 10^{8} {\rm m}$.