SHM Velocity

SHM Velocity

Consider a body handing on a spring and oscillating. The displacement x of the body from the rest position at a time t may be given by

x = x₀ sin ωt.

where x₀ is the amplitude, and ω is the angular frequency. The angular frequency is related to the period T by

ω = 2π/T.

sin ωt changes as t increases, It is zero when t is zero. As t increases, sin ωt alternates between positive and negative values. This means that when t is zero, the body just passes through the rest position. Then it moves above the rest position, if we assume that up is the positive direction. It reaches the highest position, which is at a distance of x₀ above the rest position. Then it comes back down and passes through the rest position again. It reaches the lowest position, which is at a distance of x₀ below the rest position. Then it moves back up. When it reaches the rest position, it completes one cycle. Then it repeats the cycle.

Note that this formula assumes that at the start time t = 0, the body is just passing the rest position where x = 0. We can of course start timing at any point of the motion, but we shall just use this as an example to make things simple.

The velocity is given by this formula:

v = v₀ cos ωt.

If we look at the graph of cos ω, it looks nearly the same way as the graph of sin ωt. The difference is that when t = 0, cos ωt is 1 and not 0.

This means that when the body just passes the rest position, its velocity is at a maximum. When the body is at its furthest from the rest position, the velocity at that instant is zero.

Recall that we can associate a circular motion with and oscillatory motion. A body moving round a circle can always keep the same height as a body oscillating on a spring. This is if the radius ofthe circle is equal to the amplitude x₀, and the angular velocity is equal to ω.

When the body on the circle passes the level of the rest position, it velocity is in the same direction as the body on the spring. This means that they are equal. At the rest position, the speed on the spring is at its maximum, v₀. So this is also the speed on the circle. We know that the rotational speed is given by

v = ωr,

where r is the radius. So in this case, we have the relation

v₀ = ω x₀.

Substituting into the velocity formula, we find

v = ω x₀ cos ωt.

Multiplying the displacement formula by ω, we get

ω x = ω x₀ sin ωt.

Squaring and adding, we find

v² + ω² x² = ω² x₀².

I have made use of the identity

sin² ωt + cos² ωt = 1.

This can be rearranged to give a formula for v in terms of x:

v = ± ω √( x₀² - x²).

The Physics Notes