We can also get this result from the work-energy theorem. Since only two forces are acting on the object—gravity and the normal force—and the normal force doesn’t do any work, the net work is just the work done by gravity. This only depends on the object’s weight and the difference in height, so
where
y is positive up. The work-energy theorem says that this equals the change in kinetic energy:
Using a right triangle, we can see that
so the result for the final speed is the same.
What is gained by using the work-energy theorem? The answer is that for a frictionless plane surface, not much. However, Newton’s second law is easy to solve only for this particular case, whereas the work-energy theorem gives the final speed for any shaped frictionless surface. For an arbitrary curved surface, the normal force is not constant, and Newton’s second law may be difficult or impossible to solve analytically. Constant or not, for motion along a surface, the normal force never does any work, because it’s perpendicular to the displacement. A calculation using the work-energy theorem avoids this difficulty and applies to more general situations.
Problem-solving strategy: work-energy theorem
Draw a free-body diagram for each force on the object.
Determine whether or not each force does work over the displacement in the diagram. Be sure to keep any positive or negative signs in the work done.
Add up the total amount of work done by each force.
Set this total work equal to the change in kinetic energy and solve for any unknown parameter.
Check your answers. If the object is traveling at a constant speed or zero acceleration, the total work done should be zero and match the change in kinetic energy. If the total work is positive, the object must have sped up or increased kinetic energy. If the total work is negative, the object must have slowed down or decreased kinetic energy.
Loop-the-loop
The frictionless track for a toy car includes a
loop-the-loop of radius
R . How high, measured from the bottom of the loop, must the car be placed to start from rest on the approaching section of track and go all the way around the loop?
A frictionless track for a toy car has a loop-the-loop in it. How high must the car start so that it can go around the loop without falling off?
Strategy
The free-body diagram at the final position of the object is drawn in
[link] . The gravitational work is the only work done over the displacement that is not zero. Since the weight points in the same direction as the net vertical displacement, the total work done by the gravitational force is positive. From the work-energy theorem, the starting height determines the speed of the car at the top of the loop,
where the notation is shown in the accompanying figure. At the top of the loop, the normal force and gravity are both down and the acceleration is centripetal, so
The condition for maintaining contact with the track is that there must be some normal force, however slight; that is,
. Substituting for
and
N , we can find the condition for
.
Solution
Implement the steps in the strategy to arrive at the desired result:
Significance
On the surface of the loop, the normal component of gravity and the normal contact force must provide the centripetal acceleration of the car going around the loop. The tangential component of gravity slows down or speeds up the car. A child would find out how high to start the car by trial and error, but now that you know the work-energy theorem, you can predict the minimum height (as well as other more useful results) from physical principles. By using the work-energy theorem, you did not have to solve a differential equation to determine the height.
