Simple Harmonic Motion Simple Harmonic Motion SHM occurs

Simple Harmonic Motion

Simple Harmonic Motion SHM occurs when there is an oscillation about a fixed position. A force, F, acts to return the object to its rest position and this force is directly proportional to the displacement, y, from the fixed position.

1) Rest (fixed) position. 2) Oscillation has now started. F y

3) Change of direction F y The force always acts in the opposite direction to the displacement.

Simple Harmonic Motion displacement time graph y Amplitude, A t

Spring Constant As F is directly proportional to y, but acts in the opposite direction: F = - ky where k is the spring constant, measured in Nm-1.

As F = ma ma = - ky m Therefore: d 2 y = - ky dt 2 m

The equation on the previous page leads us to: d 2 y = - ω2 y dt 2 Where ω2 is the constant of proportionality and ω is the angular frequency of motion

Also: f=1 T ω = 2 Πf ω = 2π T

SHM – Equations of Motion One end of a thin flexible rod vibrates vertically, performing SHM with a frequency, f. The displacement, y, of the end of the rod is: y

Now, compare this motion with that of an object moving in a circle. t=t A θ ω y t=0 From this, it can be seen that: y = A sin θ As θ = ωt: y = A sin ωt This basic equation describes SHM.

An object moving in such a way that its displacement with time is given by y = A sinωt or y = A cosωt is said to be executing simple harmonic motion (SHM) about the origin. The proof of this is shown below.

y = A sin (ωt) y = A cos (ωt) Differentiate both twice with respect to time: dy = Aω cos (ωt) dt dy = - Aω sin (ωt) dt d 2 y = - Aω2 sin (ωt) dt 2 d 2 y = - Aω2 cos (ωt) dt 2 d 2 y = - ω2 y dt 2 As this is the basic equation for SHM, it proves that both equations describe SHM.

Note: y = A sin(ωt) is used when y = 0, at t = 0. y = A cos(ωt) is used when y = A, at t = 0.