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) E = ½ k A 2 At equilibrium, all energy is KE and v o is the max velocity E = ½ m v o 2

Algebraic Manipulation ½ mv 2 + ½ kx 2 = ½ k A 2

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 = 2  r/ T At max displacement r = A v o = 2  A/T T = 2  A/ v o ½ kA 2 = ½ mv o 2 A/v o =  (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?