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Understanding Energy in Simple Harmonic Motion and Spring Mechanics

This comprehensive guide explores the principles of energy in simple harmonic motion (SHM) and spring mechanics. We delve into concepts such as potential energy (PE), kinetic energy (KE), and total energy in oscillating systems. The guide includes mathematical formulations, examples involving spring constants and mass, and calculations of maximum velocity and acceleration. Furthermore, we discuss the period of vibration for oscillating springs and pendulums, along with examples illustrating how these concepts apply to real-world scenarios, such as grandfather clocks.

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Understanding Energy in Simple Harmonic Motion and Spring Mechanics

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  1. Energy And SHM

  2. Energy of Spring • Spring has elastic potential energy PE = ½ kx2 • If assuming no friction, the total energy at any point is the sum of its KE and PE E = ½ mv2 + ½ kx2

  3. At Extreme • Stops moving before starts back, so all energy is PE and x is max extension (A) E = ½ k A2 • At equilibrium, all energy is KE and vo is the max velocity E = ½ m vo2

  4. Algebraic Manipulation • ½ mv2 + ½ kx2 = ½ k A2

  5. Example • If a spring is stretched to 2A what happens to a) the energy of the system? B) maximum velocity? C) maximum acceleration?

  6. Example • A spring stretches .150m when a .300kg mass is hung from it. The spring is stretched and additional .100m from its equilibrium point then released. Determine a) k b) the amplitude c) the max velocity d) the velocity when .050 m from equilibrium e) the max acceleration f) the total energy

  7. Period of Vibration • Time for one oscillation depends on the stiffness of the spring • Does not depend on the A • SHM can be thought of similar to an object moving around a circle • Time for one oscillation is the time for one revolution

  8. v = 2r/ T • At max displacement r = A • vo = 2A/T • T= 2A/ vo • ½ kA2 = ½ mvo2 • A/vo = (m/k) • T = 2(m/k)

  9. Period of oscillation depends on m and k but not on the amplitude • Greater mass means more inertia so a slower response time and longer period • Greater k means more force required, more force causes greater acceleration and shorter period

  10. Example • How long will it take an oscillating spring (k = 25 N/m) to make one complete cycle when: a) a 10g mass is attached b) a 100g mass is attached

  11. Pendulum • Small object (the bob) suspended from the end of a lightweight cord • Motion of pendulum very close to SHM if the amplitude of oscillation is fairly small • Restoring force is the component of the bobs weight – depends on the weight and the angle

  12. Period of Pendulum • T = 2√(L/ g) • Period does not depend on the mass • Period does not depend on the amplitude

  13. Example • Estimate the length of the pendulum in a grandfather clock that ticks once every second. B) what would the period of a clock with a 1.0m length be?

  14. Example • Will a grandfather clock keep the same time everywhere? What will a clock be off if taken to the moon where gravity is 1/6 that of the earth’s?

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