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Simple Harmonic Motion & Elasticity

Learn about Elastic Potential Energy in materials resulting from compression or stretching, Hooke's Law, Force-Displacement relationship, and oscillatory motion in springs and pendulums.

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Simple Harmonic Motion & Elasticity

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  1. Simple Harmonic Motion & Elasticity Chapter 10

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

  3. 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

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

  5. 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 .

  6. 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 =

  7. 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: =

  8. Felastic Fg Example 2: • 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? • 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 = PEelastic = PEelastic =

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

  10. 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.

  11. 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.

  12. 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?

  13. 10.3 Energy and Simple Harmonic Motion

  14. 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)

  15. 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).

  16. 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.

  17. 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 , = .

  18. 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.

  19. 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 .

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