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Energy And SHM 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 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 Algebraic Manipulation ½ mv2 + ½ kx2 = ½ k A2 Example If a spring is stretched to 2A what happens to a) the energy of the system? B) maximum velocity? C) maximum acceleration? 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 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 v = 2r/ T At max displacement r = A vo = 2A/T T = 2A/ vo ½ kA2 = ½ mvo2 A/vo = (m/k) T = 2(m/k) 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 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 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 Period of Pendulum T = 2√(L/ g) Period does not depend on the mass Period does not depend on the amplitude 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? 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?