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Simple Harmonic Motion

Simple Harmonic Motion. Chapter 12 Section 1. Periodic Motion. A repeated motion is what describes Periodic Motion Examples: Swinging on a playground swing Pendulum of a clock Wrecking ball Springs and oscillators Etc…. Mass-Spring System. -x. +x. F elastic. Maximum Displacement.

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Simple Harmonic Motion

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  1. Simple Harmonic Motion Chapter 12 Section 1

  2. Periodic Motion • A repeated motion is what describes Periodic Motion • Examples: • Swinging on a playground swing • Pendulum of a clock • Wrecking ball • Springs and oscillators • Etc…

  3. Mass-Spring System -x +x Felastic Maximum Displacement Felastic = 0 Equilibrium Felastic Maximum Displacement x = 0

  4. Velocity In Periodic Motion • At the equilibrium position, the mass in periodic motion reaches its maximum velocity. • At the maximum displacement away from the equilibrium position, the velocity will be zero.

  5. Acceleration In Periodic Motion • The acceleration of a mass is greatest at the maximum displacement away from the equilibrium position. • The acceleration of a mass is zero at the equilibrium position.

  6. Graphing Position, Velocity & Acceleration as a Function of Time Position (m) / Velocity (m/s) / Accel (m/s2) Time (sec)

  7. Forces In Periodic Motion • According to Newton’s Laws of Motion if there is a net force acting on an object, the object will acceleration in the direction of the net force. • The acceleration of the mass is greatest when the force is greatest. • This is when the mass reaches the maximum distance away from the equilibrium position. • The direction of the spring force will always point towards the equilibrium position.

  8. Ideal Periodic Motion • In an ideal system the mass in a periodic motion will oscillate indefinitely. • But, there is always friction present in the physics world which slows down the motion. • This slowing down motion is called damping. • Damping is minimal during short periods of time, so it can be considered an ideal system.

  9. Restoring Forces • The forces acting on the object will always pull it towards the equilibrium position. • This is why it is sometimes called the Restoring Force. • The restoring force is proportional to the displacement.

  10. Simple Harmonic Motion • Simple Harmonic Motion – Vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium. • This can be seen in most springs.

  11. Hooke’s Law • In the case of a mass-spring system, the relationship between force and displacement, discovered by Robert Hooke in 1678, is known as Hooke’s Law. Felastic = -kx Spring Force = -(Spring Constant) (Displacement)

  12. Explanation of Hooke’s Law • The negative sign is needed because the direction of the spring force is always opposite the direction of the displacement. • Spring Constant – “k” – Is a measure of the springs stiffness. • Larger the “k”, the more force is needed to stretch the spring. • SI units – N/m • The law only holds true if it doesn’t pass the springs elastic limit.

  13. Example Problem • A 76N crate is attached to a spring that has a spring constant of 450N/m. How much displacement is caused by the weight of this crate?

  14. Example Problem Answer • -0.17m

  15. Energy Within a Spring • A stretched or compressed spring has elastic potential energy. • Once the mass on the spring starts to move, the potential energy is transformed to kinetic energy. • The total mechanical energy remains constant through out the motion of the mass.

  16. Elastic Potential Energy

  17. Graphing PE, KE & ME as a Function of Time KE (J) / PE (J) / ME (J) Time (sec)

  18. Bob All the mass of the bob is concentrated at a point (center of mass) Fixed string Top attached to a fixed position and the bottom end attached to bob There is no air resistance and the mass of the string is negligible. The Simple Pendulum θ

  19. Restoring Force of a Pendulum • The restoring force of a pendulum is a component of the bob’s weight • Perpendicular to the string. • The perpendicular component of the weight force is what pulls the bob towards the equilibrium position. • Hence the restoring force.

  20. Force Diagram of a Pendulum θ

  21. Simple Harmonic Motion of a Pendulum • A pendulum is a simple harmonic oscillator as long as the angle of displacement is small. • Less than 15 degrees • At any displacement, a simple pendulum has gravitational potential energy. • Mechanical Energy is conserved. • Potential energy is from gravity, not elastic like a mass-spring system. • Graphs of KE and PE are exactly the same for both the pendulum and mass-spring system.

  22. Simple Harmonic Motion Diagram

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