Elasticity, Oscillations, Waves, and Sound
Hooke's Law
Elastic – a material is elastic if once distorted and then released, it returns to its original configuration.
Hooke's Law – maintains that for a linearly elastic object, the distortion resulting from an applied force is directly proportional to that force.
(1) 
Slope of each curve is the springs elastic constant, k.
Example 1
A hand exerciser utilizes a coiled spring. A force of 89.0 N is required to compress the spring by 0.0191 m. Determine the force needed to compress the spring by 0.0508 m.
Elastic Potential Energy
(2) 
Mechanical Energy
(3) 
Example 2
An object
of mass m = 0.200 kg is vibrating on a horizontal frictionless table.The
spring has a spring constant k = 545 N/m. It is stretched initially
to 
and
then released from rest. Determine the final translational speed 
of
the object when the final displacement of the spring is (a) 
and (b) 
.
Simple Harmonic Motion
SHM occurs when the net force along the direction of motion is a Hooke's Law type of force, that is, when the net force is proportional to the displacement and in the opposite direction.
Period (T)– the time it takes a system to progress through one complete cycle returning to its original state.
Frequency (f)– the number of cycles per second.
(4) 
or
(5) 
Angular Frequency – number of radians per second.
(6) 
Amplitude – The maximum displacement from equilibrium (A).
Displacement in SHM
(7) 
Velocity in SHM
(8) 
Acceleration in SHM
(9) 
Example 3
An air-track cart attached to a spring completes one oscillation every 2.4 s. At t = 0 the cart is released from rest at a distance of 0.10 m from its equilibrium position. What is the position of the cart at (a) 0.30 s, (b) 0.60 s, (c) 2.7 s, and (d) 3.0 s?
Example 4
The air-track cart of the previous example is used for this example. What are the velocity and acceleration of the cart at (a) 0.30 s, (b) 0.60 s?
Example 5
On December 29, 1997, a United Airlines flight from Tokyo to Honolulu was hit with severe turbulence 31 minutes after takeoff. Data from the airplane's "black box" indicated the 747 moved up and down with an amplitude of 30.0 m and a maximum acceleration of 1.8g. Treating the up-down motion of the plane as simple harmonic, find (a) the time required for one complete oscillation and (b) the plane's maximum vertical speed.
Example 6
A red delicious apple floats in a barrel of water. If you lift the apple 2.0 cm above its floating level and release it, it bobs up and down with a period of T = 0.750 s. Assuming the motion is simple harmonic, find the position, velocity, and acceleration of the apple at the times (a) T/4 and (b) T/2.
Elastic Restoring Force
Natural Angular Frequency
(10) 
Natural Linear Frequency
(11) 
Period
(12) 
Example 7
A mass of 0.22 kg on an air-track cart is attached to a spring, it oscillates with a period of 0.84 s. What is the force constant for this spring?
Example 8
A 0.120 kg mass attached to a spring oscillates with an amplitude of 0.0750 m and a maximum speed of 0.524 m/s. Find (a) the force constant and (b) the period of motion.
Example 9
When a 0.420
kg mass is attached to a spring, it oscillates with a period of 0.350
s. If, instead, a different mass, 
,
is attached to the same spring, it oscillates with a period of 0.700
s. Find (a) the force constant of the spring and (b) the mass .
Example 10
A 0.260 kg mass is attached to a vertical spring. When the mass is put into motion, its period is 1.12 s. How much does the mass stretch the spring when it is at rest in its equilibrium position?
Example 11
A 0.980 kg block slides on a frictionless, horizontal surface with a speed of 1.32 m/s. The block encounters an unstretched spring with a force constant of 245 N/m. (a) How far is the spring compressed before the block comes to rest? (b) How long is the block in contact with the spring before it comes to rest?
Example 12
A bullet
of mass m embeds itself in a block of mass M, which
is attached to a spring of force constant k. If the initial
speed of the bullet is 
,
find (a) the maximum compression of the spring and (b) the
time for the bullet-block system to come to rest.
The Pendulum
Period and Frequency of a Pendulum
(13) 
or
(14) 
Example 13
The pendulum in a grandfather clock is designed to take one second to swing in each direction; that is, 2.00 seconds for a complete period. Find the length of a pendulum with a period of 2.00 seconds.
Example 14
If you look carefully at a grandfather clock, you will notice that the weight at the bottom of the pendulum can be moved up or down by turning a small screw. Suppose you have a grandfather clock at home that runs slow. Should you turn the adjusting screw so as to (a) raise the weight or (b) lower the weight?
Example 15
A pendulum is constructed from a string 0.627 m long attached to a mass of 0.250 kg. When set in motion, the pendulum completes one oscillation every 1.59 s. If the pendulum is held at rest and the string is cut, how long will it take for the mass to fall through a distance of 1.00 m?
Damping, Forcing and Resonance
Damping is when mechanical energy is lost to other forms, such as heat or sound, eventually coming to rest at the equilibrium position.
Amplitude Equation for Damped SHM
(15) 
Forced Oscillation
If an oscillating system is driven by an external force, it is possible for energy to be added to the system.
The natural frequency of an oscillating system is the frequency at which it oscillates when free from external distrubances.
Resonance
Resonance is the response of an oscillating system to a driving force of the appropriate frequency.
