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 *E _{tot}*=

__Example 1__: Force due to a Spring

Potential Energy *PE* = (1/2)*kx*^{2}

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 *E _{tot}* 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* E _{tot}* 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)]