University Physics I - Classical Mechanics, 2019

5.5. IN SUMMARY 101
gravitational potential energy, and kinetic energy (diagram (a)). Note that we could write the total
kinetic energy as K cm + K conv , as we did in the previous chapter, but because of the large mass
of the earth, the center of mass of the system is essentially the center of the earth, which, in our
earth-bound coordinate system, does not move at all, so K cm is, to an excellent approximation,
zero. Then, the reduced mass of the system, μ = m b M E /(m b + M E ) is, also to an excellent
approximation, just equal to the mass of the ball, so K conv = 1 2 μv2 12 = 1 2 m b(v b − v e ) 2 = 1 2 m bvb
2
(again, because the earth does not move). So all the kinetic energy that we have is the kinetic
energy of the ball, and it is all, in principle, convertible (as you can see if you replace the ball, for
instance, with a bean bag).
As the ball falls, gravitational potential energy is being converted into kinetic energy, and the ball
speeds up. As it is about to hit the ground (diagram (b)), the potential energy is zero and the kinetic
energy is maximum. During the collision with the ground, all the kinetic energy is temporarily
converted into other forms of energy, which are essentially elastic energy of deformation (like the
energy in a spring) and some thermal energy (diagram (c)). When it bounces back, its kinetic
energy will only be a fraction e 2 of what it had before the collision (where e is the coefficient of
restitution). This kinetic energy is all converted into gravitational potential energy as the ball
reaches the top of its bounce (diagram (d)). Note there is more dissipated energy in diagram (d)
than in (c); this is because I have assumed that dissipation of energy takes place both during the
compression and the subsequent expansion of the ball.
5.5 In summary
1. For conservative interactions one can define a potential energy U, such that that in the course
of the interaction the total mechanical energy E = U + K of the system remains constant,
even as K and U separately change. The function U is a measure of the energy stored in the
configuration of the system, that is, the relative position of all its parts.
2. The potential energy function for a system of two particles must be a function of their relative
position only: U(x 1 − x 2 ). However, if one of the objects is very massive, so it does not move
during the interaction, its position may be taken to be the origin of coordinates, and U written
as a function of the lighter object’s coordinate alone.
3. For a system formed by the earth and an object of mass m at a height y above the ground,
the gravitational potential energy can be written as U G = mgy (approximately, as long as y
is much smaller than the radius of the earth).
4. The elastic potential energy stored in an ideal spring of spring constant k and relaxed length
x 0 , when stretched or compressed to an actual length x, isU spr = 1 2 k(x − x 0) 2 .
5. For an object in one dimension, with position coordinate x, which is part of a system with
potential energy U(x), the motion can be predicted from the “energy landscape” formed by