Q7.4 Visualising SHM behaviour (S)

An object oscillates according to the equation
x=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.
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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

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Solution

  • The period follows from
    \[ T=\frac{2\pi}{\omega} = \frac{2\pi}{3\pi} =2/3 s \]
    while the frequency is
    \[ f=\frac{1}{T}= \frac{\omega} {2\pi} =3/2 \,Hz \]
  • The displacement
    x=6cos(3πt+π/3)
    gives x(t=0)=3.
  • Differentiating with respect to time once we get
    \[ v = \dot{x} = -18 \pi sin(3\pi t +\pi/3) \]
    so that v(t=0)=-48.9m/s.
  • Differentiating once more gives the acceleration which can be written as

    a=-ω2x
    Thus 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:

Graphic - No title - vistuta
Graphic - No title - vistutb
Graphic - No title - vistutc