Oscillations Energies of S.H.M

Kinetic Energy Recall that the velocity of a particle vibrating with simple harmonic motion varies with time and, consequently, with the displacement of the particle. For the case where displacement x is zero at time t = 0, displacement and velocity are given by 𝑥= 𝑥 0 sin 𝜔𝑡 𝑣= 𝑥 𝑜 𝜔 cos ω𝑡

Relation Using the trigonometrical relation 𝑠𝑖𝑛 2 𝜃+ 𝑐𝑜𝑠 2 𝜃=1 Relating the two equations using this we get 𝑣= ±𝜔 ( 𝑥 0 2 − 𝑥 2 )

Kinetic Energy The kinetic energy of the particle (of mass m) oscillating with S.H.M is 1 2 𝑚 𝑣 2 , so the kinetic energy 𝐸 𝑘 at displacement x is given by 𝐸 𝑘 = 1 2 𝑚 𝜔 2 ( 𝑥 0 2 − 𝑥 2 )

Potential Energy The restoring force at displacement x is 𝐹 𝑟𝑒𝑠 =−𝑚 𝜔 2 𝑥 To find the change in potential energy, we need to find the work done against the restoring force 𝐸 𝑝 = 1 2 𝑚 𝜔 2 𝑥 2

Total Energy Total Energy is given by adding kinetic and potential energies together 𝐸 𝑡𝑜𝑡 = 1 2 𝑚 𝜔 2 𝑥 0 2

Example

Classwork