Q7.4 Visualising SHM behaviour (S)
An object oscillates according to the equationx=xmcos(ωt+φ)
where x describes the displacement of the object at time t,
xm=6 m, ω=3π rad/s and φ=π/3 rad.
Find the period and the frequency. Sketch
the displacement, velocity and acceleration as functions of t. Find their
values at t=0 and t=2 s.Hint
Reveal
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Hint
You will need to generate the expression for the velocity by differentiation. The acceleration follows by differentiating again, or by appeal to the SHM equation.Solution
Reveal
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Solution
- The period follows from
while the frequency is
- The displacement
x=6cos(3πt+π/3)gives x(t=0)=3.
- Differentiating with respect to
time once we get
so that v(t=0)=-48.9m/s.
Differentiating once more gives the acceleration which can be written as
a=-ω2xThus a(t=0)=-266m/s2. In t=2s the system completes t/T=3 complete cycles; so everything (displacement, velocity and acceleration) is restored to its initial value.Sketching the three functions out gives the following: