PHYSICS 201: ENERGY PLOTS FOR CONSERVATIVE FORCES
A plot of potential energy and total energy for an object can be a very useful tool in analyzing the object's motion. If only conservative forces are doing work, then we can write Etot= KE + å PE and, by the Law of Conservation of Energy, we can write KEi+å PEi=KEf+å PEf. Note that there is a potential energy for each conservative force acting on the object.
Example 1: Force due to a Spring
Potential Energy PE = (1/2)kx2
Consider the plot of potential energy shown below for a spring which is attached to a vertical wall so that the spring is aligned in the horizontal direction.
(a) What is the equilibrium position of the spring? [x=0]
(b) What is the spring constant of the spring? [8 N/m]
A 0.5 kg block is attached to the spring and the block is resting on a frictionless table. The block is pulled to x=1.5 m and then released.
(c) What is the total energy of the block? Plot Etot on the graph. [9 J, see plot]
(d) By examining the graph, find how far to the left the block will travel. [-1.5 m]
(e) At what position is the kinetic energy of the block the greatest? [x=0]
(f) Find the potential and kinetic energy of the block at x=1 m. [PE = 4J, KE = 5 J]
(g) Find the speed of the block at x=1 m. [4.47 m/s]
(h) In which direction is the force on the block at x=1 m? [to the left]
(i) In which direction is the block moving at x=1 m? at x=-1 m? [cannot tell]
(j) Is the force on the object ever zero? Where? [Yes, at x=0]
Example 2: Gravity Near Earth's Surface
Potential Energy PE = mgh
Consider a 2 kg stone that is thrown up from the ground with an initial speed of 10 m/s. The potential energy of the stone is plotted below. Note that the ground is assigned h=0. Note that we must neglect air resistance.
(a) What is the total energy of the stone? Plot Etot on the graph. [100 J, see plot]
(b) By examining the graph, find how high the stone travels. [5 m]
(c) Find the potential energy and kinetic energy of the stone at h=3 m. [PE ~ 60 J, KE ~ 40 J]
(d) What is the stones speed at h=3 m? [~6.3 m/s]
(e) What is the direction of the force at h=3 m? [down (negative direction)]
(f) Does the force on the stone ever change direction? [No (slope never changes sign)]
(g) Is the force on the stone ever zero? [No (slope is never zero)]