Simple Harmonic Motion & Elasticity Chapter 10

Elastic Potential Energy What is it? Energy that is in materials as a result of their . Where is it found?

Law A spring can be or with a . The by which a spring is compressed or stretched is to the magnitude of the ( ). Hooke’s Law: Felastic = Where: = spring constant = of spring ( ) = displacement

Hooke’s Law What is the graphical relationship between the elastic spring force and displacement? Felastic = -kx Displacement Force

Hooke’s Law A force acting on a spring, whether stretching or compressing, is always . Since the spring would prefer to be in a “relaxed” position, a negative “ ” force will exist whenever it is deformed. The force will always attempt to bring the spring and any object attached to it back to the position. Hence, the restoring force is always .

Example 1: A 0.55 kg mass is attached to a vertical spring. If the spring is stretched 2.0 cm from its original position, what is the spring constant? Known: m = x = g = Equations: Fnet = = + (1) = (2) = (3) Substituting 2 and 3 into 1 yields: k =

Elastic in a Spring The exerted to put a spring in tension or compression can be used to do . Hence the spring will have Elastic . Analogous to kinetic energy: =

Example 2: A 0.55 kg mass is attached to a vertical spring with a spring constant of 270 N/m. If the spring is stretched 4.0 cm from its original position, what is the Elastic Potential Energy? Known: m = 0.55 kg x = -4.0 cm k = 270 N/m g = 9.81 m/s2 Equations: PEelastic = What is the maximum value of elastic potential energy of the system when the spring is allowed to oscillate from its relaxed position with no weight on it? Fg Felastic

Elastic Potential Energy What is area under the curve? A = Which you should see equals the Displacement Force

Simple Harmonic Motion & Springs An around an will occur when an object is from its equilibrium position and . For a spring, the restoring force F = -kx. The spring is at equilibrium when it is at its relaxed length. ( ) Otherwise, when in tension or compression, a restoring force exist.

Simple Harmonic Motion & Springs At displacement (+ ): The Elastic Potential Energy will be at a The force will be at a . The acceleration will be at a . At (x = ): The Elastic Potential Energy will be Velocity will be at a . Kinetic Energy will be at a The acceleration will be , as will the force.

10.3 Energy and Simple Harmonic Motion Example 3 Changing the Mass of a Simple Harmonic Oscilator A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. When released from rest, how far does the ball fall before being brought to a momentary stop by the spring?

10.3 Energy and Simple Harmonic Motion

Simple Harmonic Motion of Springs Oscillating systems such as that of a spring follow a pattern. Harmonic Motion of Springs – 1 Harmonic Motion of Springs (Concept Simulator)

Frequency of Oscillation For a spring oscillating system, the frequency and period of oscillation can be represented by the following equations: Therefore, if the of the spring and the are known, we can find the and at which the spring will oscillate. k and mass equals frequency of oscillation (A spring).

Harmonic Motion & The Simple Pendulum Simple Pendulum: Consists of a massive object called a suspended by a string. Like a spring, pendulums go through as follows. Where: = Note: This formula is true for only of . The period of a pendulum is of its mass.

Conservation of ME & The Pendulum In a pendulum, is converted into and vise-versa in a continuous repeating pattern. PE = mgh KE = ½ mv2 MET = PE + KE MET = Note: kinetic energy is achieved at the point of the pendulum swing. The potential energy is achieved at the of the swing. When is , = , and when is , = .

Key Ideas Elastic Potential Energy is the in a spring or other elastic material. Hooke’s Law: The of a spring from its is the applied. The of a vs. is equal to the . The under a vs. is equal to the done to compress or stretch a spring.

Key Ideas Springs and pendulums will go through oscillatory motion when from an position. The of of a simple pendulum is of its of displacement (small angles) and . Conservation of energy: Energy can be converted from one form to another, but it is .