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Illustration 7.5: A Block on an Incline
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A block is on an incline and slides without friction. Partway down the incline, it hits a spring as shown (position is given in meters, time is given in seconds, and energy on the bar graph is given in joules). You can add the protractor by checking the box. Also shown are the force vectors, one for each force (the red ones) and one for the total force acting (the blue one). The energy of the system is shown in the three bar graphs on the right: kinetic energy (orange), gravitational potential energy (blue), and elastic potential energy (green). Restart.
Let's begin by analyzing this situation as we would have in Chapters 3 and 4. First, we need to define a convenient set of axes. A convenient set of axes has one axis along the incline and the other axis perpendicular to the incline. This choice allows us to have one direction where there is no acceleration (the direction perpendicular to the incline) and one direction where there is an acceleration (parallel to the incline). There is also another reason for this choice of axes. It allows us to decompose only one force instead of two. We have to decompose the gravitational force into a component along the incline and one perpendicular to the incline. How do we deal with the spring force? Well, the honest answer is that while we can analyze the forces to determine the acceleration, it is not tremendously useful since the spring force is not constant.
Run the animation and look at the normal force and the gravitational force vs. the spring force. The spring is not compressed initially, then it compresses, and then it uncompresses. During this time the net force on the block changes dramatically. Look at the blue net force vector. As a consequence, the acceleration of the block changes dramatically as well. (Note that the net force still points parallel to the incline; its size is what changes dramatically.)
Since the forces change over the course of the motion of the block, the acceleration of the block is not constant throughout the motion of the block. Newton's laws and kinematics clearly fall short in analyzing the motion here. What to do? Use energy! At the starting point of the motion of the block, it has no kinetic energy, and no elastic (spring) potential energy, but it does have gravitational potential energy. As the block moves down the incline some of the gravitational potential energy is converted to kinetic energy. When the block hits the spring, the kinetic energy and the gravitational potential energy get converted to elastic (spring) potential energy.
Watch the animation and describe how all of the potential energy due to the compressed spring gets converted to other types of energy.
Illustration authored by Mario Belloni.
Script authored by Steve Mellema and Chuck Niederriter and modified by Mario Belloni.