When spring is stretched or compressed it has elastic potential energy.

Welastic = 1/2 kx02 - 1/2 kxf

2

or Initial elastic potential energy minus Final elastic potential energy.

PEelastic = 1/2 kx2 where k is the spring constant, and x is the distance the spring is compressed or stretched beyond its unstrained length. The unit is the joule (J).

When external nonconservative forces do no net work on a system then total mechanical energy must be conserved.

Ef = E0

Total mechanical energy = translational kinetic energy + rotational kinetic energy + gravitational potential energy + elastic potential energy.

If there is no rotation, this becomes this equation:

E = 1/2 mv2 + mgh + 1/2 kx2

Ex. 10 - A m = 0.200 kg object is vibrating on a horizontal, frictionless table. The spring (k = 545 N/m) is stretched initially to x0 = 0.0450 m and then released from rest. Determine the final translational speed vf

of the object when the final displacement of the spring is (a) xf = 0.0255 m and (b) xf = 0.

Ex. 12 - A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. The ball, supported initially so that the spring is neither stretched nor compressed, is released form rest. How far does the ball fall before being momentarily stopped by the spring?

A simple pendulum is a mass m suspended by a pivot P. When the object is pulled to one side and released, it will swing back and forth in a motion approximating simple harmonic motion.

A series of substitutions finds that, for small angles,

2πf = √g/L

f is frequency, g is 9.80, and L is length.

2πf = √g/LMass is algebraically eliminated, and it has no bearing on the frequency of a pendulum.

Ex. 13 - Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of 1.00 s.

A pendulum can be a real object, in which case it is called a physical pendulum.

In reality, an object in simple harmonic motion will not vibrate forever. Friction, or some such force, will decrease the velocity and amplitude of the motion. This is called damped harmonic motion.

A shock absorber introduces damping forces to reduce the vibrations of the ride. The smallest degree of damping that completely eliminates the oscillations is called critical damping.

When damping exceeds the critical value, this is called overdamping. Underdamping is when the damping forces are less than what would eliminate the vibration.

Automobile shocks are generally designed to produce underdamped motion.

In driven harmonic motion, a driving force is applied to assist the object in its vibrational motion.

If the frequency of this driving force is the same as the natural frequency of the system, the total mechanical energy of the system increases.

Resonance is when a periodic force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion.

Resonance usually occurs when the frequency of the driving force is equal to the natural frequency of the object.

Formulas for frequency and period of an oscillating spring:

2•π•f = √ k/m2•π/ T = √ k/m