A Ball on an Incline

(Illustration)

check, then click the parameters and play button to make the spring invisible.  
check, then click set parameters and play button to add the angle tool.

Please wait for the animation to completely load.

A ball is on an incline and slides without friction until it hits a spring as shown (position is given in meters and time is given in seconds).  You can make the spring invisible and add the angle tool by checking one of the above boxes.  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 going about this problem as we would have in Chapters 3 and 4.  First we need to define a convenient set of axes.  A convenient set of axes is with 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 (down the incline).  There is also another reason for this choice.  It allows use 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 as the spring force is not constant.

Run the animation and look a the Normal force and the gravitational force verses the spring force.  The spring is not compressed initially, then compresses, then uncompresses.  During this time the net force on the ball changes dramatically.  As a consequence the acceleration of the ball changes dramatically as well.

Since the forces change over the course of the motion of the ball, the acceleration of the ball is not constant throughout the motion of the ball.  Newton's laws and kinematics clearly fall short.  What to do?  Use energy!  At the starting point of the motion of the ball, it has no kinetic energy, no elastic (spring) potential energy, and gravitational potential energy.  As the ball moves down the incline some of the gravitational potential energy is converted to kinetic energy.  When the ball 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.