Question

Zontini the Amazing Hypnotist is hypnotizing a volunteer at a performance by

swinging a pocket watch back and forth in front of her face. The pocket watch's
current distance from center in centimetres in terms of time is modelled by the
equation d(s) = 7 sin(360s). If Zontini the Amazing Hypnotist were to swing the watch
twice as fast, covering the same distance, how would the equation change?

Answer 1

The equation will change by an increase in the angular frequency of motion by a factor of 2 to become d(s) = 7·sin(720s)

Explanation:

The given oscillatory motion equation of the swinging pocket watch is d(s) = 7sin(360s)

The general equation of simple harmonic motion is x = A·sin(ω₁t + ∅)

Comparing, we have;

x = d(s)

A = 7

ω₁t = 360

∅ = 0

The period of oscillation = The time to complete a cycle = 2·π/ω₁

Therefore;

ω₁ = 2·π/T₁

When the cycle or the watch swing rate is doubled, the time taken to compete one cycle is halved and the new period, T₂ = T₁/2

ω₂ = 2·π/T₂ = 2·π/(T₁/2) = 4·π/T₁ = 2ω₁

The equation becomes;

x = A·sin(ω₂t) = A·sin(2ω₁t) which gives;

d(s) = 7·sin(2 × 360s) = 7·sin(720s)

The equation will change by the doubling of the angular frequency of the motion