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Energy And SHM

Energy And SHM. Energy of Spring. Spring has elastic potential energy PE = ½ kx 2 If assuming no friction, the total energy at any point is the sum of its KE and PE E = ½ mv 2 + ½ kx 2. At Extreme. Stops moving before starts back, so all energy is PE and x is max extension (A)

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Energy And SHM

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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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