Ch 11 Notes
11.2 Energy in the Simple Harmonic Oscillator
Forces are not exactly constant in a SHO, as the object moves back and forth the force varies with as the distance from the origin varies.
Energy can therefore be used as we investigate the properties of the SHO.
That is because it takes a certain amount of energy to compress or stretch a spring. Just like the mass stretched/compressed the spring.
We will call this the Elastic Potential Energy that is stored in the spring after compression/stretching.
The greater the compression/stretch (change in “x”) the greater is the elastic potential energy.
The formula is PE = ½ kx2 where PE is the elastic potential energy in Joules, k is the spring constant in N/m, and x is the distance or displacement from the origin.
Now if the spring is let go it begins to move – Kinetic Energy is the measured energy of motion. KE = ½ mv2 where KE is the kinetic energy in Joules, m is the mass of the moving object in kg, and v squared is the velocity measured in m/s.
The moving object starts having all elastic potential energy and begins to gain kinetic energy as it starts to move.
As long as there is no friction, the Total Mechanical Energy remains constant as the sum of the kinetic and potential energies. ET = ½ mv2 + ½ kx2
As the mass oscillates back and forth, the energy continuously changes from potential energy to kinetic energy, and back again.
Do you remember amplitude “A” it is the full distance “x” that the spring was compressed or stretched. Because amplitude is the distance in which the SHO has the max elastic potential energy with no kinetic energy ½ A x2 = ½ mv2 + ½ kx2 because there is no velocity, we can simply say ½ A x2 = ½ kx2 = TE
Or we can watch as the SHO passes the resting position and say ½ mv2 = TE this would represent the maximum velocity of the mass because there would be no elastic potential energy if “x” equals zero.
If we want the velocity of the mass at any position “x” we can use the formula
So, what properties can we evaluate as an object oscillates back and forth?
1. The total elastic potential energy when completely stretched/compressed
2. The total kinetic energy as it passes the center
3. The total energy as it is somewhere between the center and the “+x” or “-x”
4. The maximum velocity through the center
5. The velocity at any point “+x” or “-x”
6. Of course we have to calculate the spring constant
7. And we can also find the maximum acceleration amax = Fmax = kA
m m
Exit Question:
Suppose a spring is stretched twice as far as the original stretched position, what happens to the;
a. energy of the system
b. maximum velocity
c. maximum acceleration