Question

A mass attached to a vertical spring has position function given by s(t)=4sin(2t)where t is measured in seconds and s in inches.

Answer 1

Given data:

The position function of spring is s(t)=4sin(2t).

By differentiating the given position function with respect to time is the velocity of the system. It can be calculated as,

`\begin{gathered} s(t)=4\sin (2t) \\ \frac{d}{\mathrm{d}t}s(t)=\frac{d}{\mathrm{d}t}(4\sin (2t)) \\ v(t)=4\cos (2t)*2 \\ v(t)=8\cos (2t) \end{gathered}`

The acceleration of the system can be calculated by differentiating above velocity function with respect to time,

`\begin{gathered} v(t)=8\cos (2t) \\ \frac{d}{\mathrm{d}t}v(t)=\frac{d}{\mathrm{d}t}(8\cos (2t)) \\ a(t)=-8\sin (2t)*2 \\ a(t)=-16\sin (2t) \end{gathered}`

The velocity at t=2 s will be,

`\begin{gathered} v(t)=8\cos (2t) \\ v(2)=8\cos (2*2) \\ v(2)=8\cos 4 \\ v(2)=-5.23\text{ m/s} \end{gathered}`

The acceleration at t=2 will be,

`\begin{gathered} a(t)=-16\sin (2t) \\ a(2)=-16\sin (2*2) \\ a(2)=-16\sin (4) \\ a(2)=-12.1m/s^2 \end{gathered}`

Thus, the velocity and acceleration at t=2 is -5.23 m/s and -12.1 m/s² respectively.