# A torsional pendulum - determine angular speed

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In summary, the torsional pendulum, when released, executes simple harmonic motion with an angular displacement given by \theta=0.2\cos\left(\pi(t-0.5)\right). To find the rate of change of the angular displacement with respect to time, we can use the derivative, which gives us \omega(t)=-0.2\pi\sin\left(\pi(t-0.5)\right). At 3.1 seconds, the angular displacement is decreasing at a rate of approximately 0.6 radians per second.
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.

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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?

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.

## 1. What is a torsional pendulum?

A torsional pendulum is a type of pendulum that consists of a suspended mass attached to a thin wire or rod. The motion of the pendulum is determined by the twisting or torsion of the wire, rather than the swinging motion of a traditional pendulum.

## 2. How is angular speed determined in a torsional pendulum?

Angular speed in a torsional pendulum is determined by the formula w = 2π/T, where w is the angular speed in radians per second and T is the period of the pendulum, or the time it takes to complete one full cycle of oscillation.

## 3. What factors affect the angular speed of a torsional pendulum?

The angular speed of a torsional pendulum is affected by various factors such as the length and diameter of the wire or rod, the mass of the suspended object, and the amplitude of the pendulum's oscillations.

## 4. How does a torsional pendulum demonstrate simple harmonic motion?

A torsional pendulum demonstrates simple harmonic motion because its motion follows a sinusoidal pattern, with the pendulum swinging back and forth in a periodic motion. This motion is characterized by a constant period and amplitude, and is governed by Hooke's law.

## 5. What practical applications does a torsional pendulum have?

Torsional pendulums have various practical applications, such as being used as a sensitive instrument to measure small forces, or as a timing device in clocks and watches. They are also used in seismology to measure earthquakes, and in research to study the properties of materials such as elasticity and viscosity.

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