# PHY C13: Energy in Oscillations

Today we’re covering:

• Types of Energy
• Kinetic
• Potential
• Total Energy

Let’s go!

What types of energy exist in Simple Harmonic Motion?

 Kinetic energy Definition Energy due to motion (change in displacement x over time t) Kinetic Energy at any point x Ek = ½ mv2 Since v = ω√(x02 – x2), ½ m[ω√(x02 – x2)]2 Ek = ½ mω2(x02 – x2) Maximum Kinetic Energy (when v = maximum, v0) ½ mv02 OR ½ mω2x02 Graph of Ek against Displacement (x) Graph of Ek against Time (t) Potential energy Definition Work done AGAINST restoring force (motion is always OPPOSITE direction of Fres)Consists of: Gravitational Potential Energy Elastic Potential Energy (for springs) Potential Energy at any point x Ep = ½ mω2x2   Because: Fres = – mω2x W = FΔx, Ep = FΔx Since Fres varies, we need to find the AVERAGE F to plug into the formula: F = ½Fres Thus, Ep = ½Fresx2 (the negative disappears since it shows the direction of Fres, & Ep is a scalar) Maximum Potential Energy (when a = maximum, x = maximum, x0) ½mω2x02 Graph of Ep against Displacement (x) Graph of Ep against Time (t) How do these values change?
Remember: in Free Oscillations & thus Simple Harmonic Motion, there is NO ENERGY LOSS.

TOTAL energy (Kinetic + Potential) = CONSTANT

Therefore,

Maximum Kinetic Energy = Maximum Potential Energy

(You can check in the table above: they have the same formula in fact!)

Etot = Ek + Ep
Etot = ½ mω2(x02 – x2) + ½ mω2x2
Etot = ½ mω2x02

OR

Etot = Ekmax = Epmax
Etot = ½ mω2x02