Sample Problem 7-9

Force , with in meters, acts on a particle, changing only the kinetic energy of the particle. How much work is done on the particle as it moves from coordinates (2 m, 3 m) to (3 m, 0 m)? Does the speed of the particle increase, decrease, or remain the same?

SOLUTION; The Key Idea here is that the force is a variable force because its component depends on the value of . Thus, we cannot use Eqs. 7-7 and 7-8 to find the work done. Instead, we must use Eq. 7-36 to integrate the force:

The positive result means that energy is transferred to the particle by force F. Thus, the kinetic energy of the particle increases, and so must its speed.

Power

A contractor wishes to lift a load of bricks from the sidewalk to the top of a building by means of a winch. We can now calculate how much work the force applied by the winch must do on the load to make the lift. The contractor, however, is much more interested in the rate at which that work is done. Will the job take 5 minutes (acceptable) or a week (unacceptable)?

The time rate at which work is done by a force is said to be the powerdue to the force. If an amount of work is done in an amount of time by a force, the average powerdue to the force during that time interval is

(7-42)

The instantaneous poweris the instantaneous time rate of doing work, which we can write as

(7-42)

Suppose we know the work done by a force as a function of time. Then to get the instantaneous power at, say, time s during the work, we would first take the time derivative of , and then evaluate the result for s.

The SI unit of power is the joule per second. This unit is used so often that it has a special name, the watt(W), after James Watt, who greatly improved the rate at which steam engines could do work. In the British system, the unit of power is the foot-pound per second. Often the horsepower is used. Some relations among these units are

And

W.

We can also express the rate at which a force does work on a particle (or particle like object) in terms of that force and the particle's velocity. For a particle that i! moving along a straight line (say, the x axis) and is acted on by a constant force / directed at some angle to that line, Eq. 7-43 becomes

,

or

Reorganizing the right side of Eq. 7-47 as the dot product , we may also write Eq. 7-47 as

For example, the truck in Fig. 7-13 exerts a force on the trailing load, which has velocity at some instant. The instantaneous power due to is the rate at which does work on the load at that instant and is given by Eqs. 7-47 and 7-48. Saying that this power is "the power of the truck" is often acceptable, but we should keep in mind what is meant: Power is the fate at which the applied force does work.

•CHECKPOINT 5: A block moVes with uniform circular motion because a cord tied tothe block is anchored at the center of a circle. Is the power due to the force on the block from the cord positive, negative, or zero?