At the bottom of the slope, all the box’s potential energy will have been converted into kinetic energy. In other words, the kinetic energy, 1 2 mv2, of the box at the bottom of the slope is equal to the potential energy, mgh, of the box at the top of the slope. Solving for v, we get:
4. What is the work done on the box by the force of gravity in bringing it to the bottom of the inclined plane? The fastest way to solve this problem is to appeal to the work-energy theorem, which tells us that the work done on an object is equal to its change in kinetic energy. At the top of the slope the box has no kinetic energy, and at the bottom of the slope its kinetic energy is equal to its potential energy at the top of the slope, mgh. So the work done on the box is:
Note that the work done is independent of how steep the inclined plane is, and is only dependent on the object’s change in height when it slides down the plane. Frictionless Inclined Planes with Pulleys Let’s bring together what we’ve learned about frictionless inclined planes and pulleys on tables into one exciting über-problem:
Assume for this problem that —that is, mass M will pull mass m up the slope. Now let’s ask those three all-important preliminary questions:Ask yourself how the system will move: Because the two masses are connected by a rope, we know that they will have the same velocity and acceleration. We also know that the tension in the rope is constant throughout its length. Because , we know that when the system is released from rest, mass M will move downward and mass m will slide up the inclined plane. Choose a coordinate system: Do the same thing here that we did with the previous pulley-on-a-table problem. Make the x-axis parallel to the rope, with the positive x direction being up for mass M and downhill for mass m, and the negative x direction being down for mass M and uphill for mass m. Make the y-axis perpendicular to the rope, with the positive y-axis being away from the inclined plane, and the negative y-axis being toward the inclined plane. Draw free-body diagrams: We’ve seen how to draw free-body diagrams for masses suspended from pulleys, and we’ve seen how to draw free-body diagrams for masses on inclined planes. All we need to do now is synthesize what we already know:
Now let’s tackle a couple of questions:
1. What is the acceleration of the masses?
2. What is the velocity of mass m after mass M has fallen a distance h?
1. What is the acceleration of the masses? First, let’s determine the net force acting on each of the masses. Applying Newton’s Second Law we get:
Adding these two equations together, we find that . Solving for a, we get: