## Check Your Understanding

No, only its magnitude can be constant; its direction must change, to be always opposite the relative displacement along the surface.

a. The spring force is the opposite direction to a compression (as it is for an extension), so the work it does is negative. b. The work done depends on the square of the displacement, which is the same for $x=\pm 6\phantom{\rule{0.2em}{0ex}}\text{cm}$, so the magnitude is 0.54 J.

## Conceptual Questions

When you push on the wall, this “feels” like work; however, there is no displacement so there is no physical work. Energy is consumed, but no energy is transferred.

If you continue to push on a wall without breaking through the wall, you continue to exert a force with no displacement, so no work is done.

Both require the same gravitational work, but the stairs allow Tarzan to take this work over a longer time interval and hence gradually exert his energy, rather than dramatically by climbing a vine.

The first particle has a kinetic energy of $4(\frac{1}{2}m{v}^{2})$ whereas the second particle has a kinetic energy of $2(\frac{1}{2}m{v}^{2}),$ so the first particle has twice the kinetic energy of the second particle.

The mower would gain energy if $\mathrm{-90}\text{\xb0}<\theta <90\text{\xb0}.$ It would lose energy if $90\text{\xb0}<\theta <270\text{\xb0}.$ The mower may also lose energy due to friction with the grass while pushing; however, we are not concerned with that energy loss for this problem.

The second marble has twice the kinetic energy of the first because kinetic energy is directly proportional to mass, like the work done by gravity.

Unless the environment is nearly frictionless, you are doing some positive work on the environment to cancel out the frictional work against you, resulting in zero total work producing a constant velocity.

Appliances are rated in terms of the energy consumed in a relatively small time interval. It does not matter how long the appliance is on, only the rate of change of energy per unit time.

The spark occurs over a relatively short time span, thereby delivering a very low amount of energy to your body.

If the force is antiparallel or points in an opposite direction to the velocity, the power expended can be negative.

## Problems

a. 772 kJ; b. 4.0 kJ; c. $1.8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-16}}\phantom{\rule{0.2em}{0ex}}\text{J}$

$W(\text{bullet})=20\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}W(\text{crate})$

## Additional Problems

a. $208\phantom{\rule{0.2em}{0ex}}\text{N}\xb7\text{m}$; b. $240\phantom{\rule{0.2em}{0ex}}\text{N}\xb7\text{m}$

a. $\text{\u2212}0.5\phantom{\rule{0.2em}{0ex}}\text{N}\xb7\text{m}$; b. $\mathrm{-0.83}\phantom{\rule{0.2em}{0ex}}\text{N}\xb7\text{m}$

## Challenge Problems

If crate goes up: a. 3.46 kJ; b. −1.89 kJ; c. −1.57 kJ; d. 0; If crate goes down: a. −0.39 kJ; b. −1.18 kJ; c. 1.57 kJ; d. 0

a. 40 hp; b. 39.8 MJ, independent of speed; c. 80 hp, 79.6 MJ at 30 m/s; d. If air resistance is proportional to speed, the car gets about 22 mpg at 34 mph and half that at twice the speed, closer to actual driving experience.