A torsional pendulum - determine angular speed

[HelpMePls2]

A torsional pendulum made by suspending a horizontal uniform metal disk by a wire from its center. If the disk is rotated and then released, it willexecute simple (angular) harmonic motion. Suppose at t seconds the angular displacement of \theta radians from the initial position is given by the equation \theta = 0.2 cos \pi(t-0.5). Determine to the nearest tenth of a radian per second, how fast the angle is changing at 3.1 sec.

 

[MarkFL]

Hello and welcome to MHB! (Wave)

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Okay, that's our canned response to new users who have posted a question without showing any work or thoughts. It serves us well, but let's consider, we are given:

\(\displaystyle \theta=0.2\cos\left(\pi(t-0.5)\right)\)

Now, we are being asked how at what rate the angular displacement $\theta$ is changing with respect to time $t$. What concept from calculus allows us to determine the instantaneous rate of change of one variable with respect to another?

 

[MarkFL]

To follow up, we can use the derivative to find the time rate of change of the angular displacement, which is typically denoted by $\omega$.

\(\displaystyle \omega(t)=\d{\theta}{t}=-0.2\pi\sin\left(\pi(t-0.5)\right)\)

Hence:

\(\displaystyle \omega(3.1)=-0.2\pi\sin\left(\pi(3.1-0.5)\right)=-0.2\pi\sin(2.6\pi)\approx-0.6\)

So, we find that at time $t=3.1\text{ s}$, the angular displacement is decreasing at a rate of about 0.6 radians per second.