Question

The motion of a mass on a spring hung vertically, where the mass oscillates up and down, can also be modeled using the rotating disk. Instead of the lights being placed horizontally along the top and pointing down, place the lights vertically and have the lights shine on the side of the rotating disk. A shadow will be produced on a nearby wall, and will move up and down. Write the equations of motion for the shadow taking the position at t = 0.0 s to be y = 0.0 m with the mass moving in the positive y-direction.

Solution

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First, it is convenient to draw a representation of the problem. Note that we can relate the angle θ\theta to the position xx using trigonometry. Then, we differentiate this expression in respect to time in order to obtain the velocity vv. Finally, we differentiate vv in respect to time to obtain the acceleration aa.

θ(t)=ωt    x(t)=Rsin(θ)=Rsin(ωt)    v(t)=dxdt=Rωcos(ωt)    a(t)=dvdt=Rω2sin(ωt)\begin{align*} \theta\left(t\right)&=\omega t \\ \implies x\left(t\right) &= R \sin\left(\theta\right) \\ &=\boxed{R\sin\left(\omega t\right)} \\ \implies v\left(t\right)&=\dfrac{dx}{dt}=\boxed{R \omega \cos\left(\omega t\right)} \\ \implies a\left(t\right)&=\dfrac{dv}{dt}=\boxed{-R \omega^2 \sin\left(\omega t\right)} \end{align*}

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