Simple Harmonic Motion Elasticity Chapter 10 Elastic Potential

Simple Harmonic Motion & Elasticity Chapter 10

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

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

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

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= k= k=

Elastic Spring ► The in a 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: ►What is the maximum value elastic spring potential ►A 0. 55 kg mass is attached to aofvertical with energy the system when the If spring is allowed to a springofconstant of 270 N/m. the spring is oscillate from its from relaxed with no weight stretched 4. 0 cm its position original position, what is on it? the Elastic Potential Energy? ► Known: m = 0. 55 kg x = -4. 0 cm k = 270 N/m g = 9. 81 m/s 2 ► Felastic Equations: PEelastic = Fg

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

Simple Harmonic Motion & Springs ► Simple Harmonic Motion: § 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 a systems such as that of a spring follow 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 at which the spring will oscillate. k and frequency of oscillation (A mass equals and 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: 1. 2. This formula is true for only The period of a pendulum is of. 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 mv 2 MET = PE + KE MET = Note: 1. 2. 3. 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.