Torsional Pendulum

Torsional Pendulum Definition

A torsional pendulum is much like a regular pendulum except that it exhibits rotational motion instead of translational motion.

See also the illustration below. To set a torsional pendulum in motion, you need to rotate the bob somewhat and let it loose. The bob will start to rotate back to its equilibrium position, after which it will 'overshoot' and rotate out of its equilibrium position in the other direction than you had rotated it originally. Its rotation will be slowed down again, and the bob will keep rotating in this oscillatory way.

Torsional Pendulum Theory

The theory behind the workings of a torsional pendulum all boils down to the fact that our string has some stiffness. That is, if we introduce a twist in our string, then it will have the tendency to un-twist if we let everything go and let the string do its thing. In fact, the string has the tendency to un-twist harder if we twist it harder, and this relationship is exactly linear.

The string in a torsional pendulum exerts a torque that is proportional to the angle through which the lower end of the string is turned while the upper end of the string remained fixed.

Now we can also see the resemblance with a regular pendulum: the rotational equivalent of force (torque) is proportional to the rotational equivalent of displacement (angle). Thus, we could question whether or not it would exhibit simple harmonic motion. Let's find out if that's the case by setting up some equations and solving them.

Torsional Pendulum Equations

To speak of a torsional pendulum, we assume that the torque exerted by the string on the bob is proportional to the angular displacement of the bob. This means that the string has a certain torsion constant, \(c,\) which is equal to the torque exerted by the string divided by the angular displacement of the bob. As you might have been able to derive from its definition, the torsion constant \(c\) is measured in \(\mathrm{\tfrac{N\,m}{rad}}\). The stiffer the string is, the higher the torsion constant is. We can now set up the following equation:

\[\vec{\tau}=-c\vec{\theta},\]

where \(\vec{\tau}\) is the torque exerted by the string and \(\vec{\theta}\) is the angular displacement of the bob. The minus sign is there to indicate that the torque exerted by the string is in the direction opposite to the angular displacement of the bob: after all, the string wants to un-twist itself. Therefore, we can say that this is a restoring torque in the same sense, that the force in a classical pendulum is a restoring force.

You may remember Newton's second law in angular form, which implies that the torque exerted on our bob is equal to the bob's moment of inertia \(I\) times its angular acceleration, \( \alpha \), so we have the following differential equation:

\begin{aligned}I\textcolor{#00b695}{\alpha} & = \textcolor{#56369f}{\tau}\\[8pt]I\textcolor{#00b695}{\frac{\mathrm{d}^2\vec{\theta}}{\mathrm{d}t^2}} &=\textcolor{#56369f}{-c\vec{\theta}},\end{aligned}

where \(t\) is time.

As our bob spins back and forth, the sense of the bob's angular displacement vector and of torque changes. Using the right-hand thumb rule, we can see that angular displacement's direction is vertically up as it twists to the right, and vertically down as it untwists (turns to the left). This tells us the direction of torque because we know it acts in the opposite direction as \( \vec{tau} = -c\vec{\theta}. \) Nevertheless, in both cases torque it is oriented vertically. Therefore, we are guaranteed that the motion occurs in the same horizontal plane. Since this is the case, we can leave aside the vector notation which is complicated and unnecessary for this case. Instead, we can continue by using a scalar differential equation where the algebraic sign keeps all the information needed to describe the pendulum motion.

\[\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}=-\frac{c}{I}\theta.\]

This looks suspiciously much like the differential equation governing the simple harmonic motion of the regular pendulum, and indeed, the solution to this equation describes simple harmonic motion:

\[\theta(t)=\theta_\text{max}\sin\left(\sqrt{\frac{c}{I}}t\right),\]

where \(\theta_\text{max}\) is the angular amplitude or the maximum angular displacement of the bob.

From this equation, we see that the angular frequency \(\omega\) is given by

\[\omega=\sqrt{\frac{c}{I}}.\]

The period \(T\) of the oscillatory motion is given by

\begin{aligned}&T=\frac{2\pi}{\omega}\\[6pt]&\boxed{T=2\pi\sqrt{\frac{I}{c}}}.\end{aligned}

Energy in a Torsional Pendulum

We can find an expression for the potential energy stored in a torsional pendulum. We can define the potential energy as minus the work done by the string on the bob from its equilibrium position to its current position. The work \(W\) done by a torque \(\vec{\tau}\) is equal to \(\int\vec{\tau}\cdot\mathrm{d}\vec{\theta}\) so the potential energy \(E_\text{pot}\) in the torsional pendulum can be expressed as follows:

\begin{align*}&E_\text{pot}=-W\\&E_\text{pot}=-\int\vec{\tau}\cdot\mathrm{d}\vec{\theta}\\&E_\text{pot}=-\int(-c\vec{\theta})\cdot\mathrm{d}\vec{\theta}\\&\boxed{E_\text{pot}=\frac{1}{2}c\theta^2}.\end{align*}

Torsional Pendulum Experiment

You can do an experiment with a torsional pendulum yourself if you have two things at your disposal: a tire swing and a windless day.

To actually have the torque exerted by the rope be proportional to the angular displacement of the tire, we should use an initial angular amplitude not larger than \(270^\circ\). That means that you should not rotate the tire more than 3/4 of a full turn from its equilibrium position. Now let go of the tire and measure the oscillation period with a stopwatch. You can best do this by measuring the time it takes for the tire to go from a stationary point to the next stationary point: this will of course be half a period.

Did you manage to measure the period of this simple harmonic motion? Good! Now, if we assume a typical value of a car tire's moment of inertia of \(I_\text{tire}=0.4\,\mathrm{kg\,m^2}\), you can calculate the torsion constant of the rope using the formula for the period of motion! If you manipulate this equation correctly, you should get

\[c=I\left(\frac{2\pi}{T}\right)^2.\]

Torsional Pendulum Clock

Fig. 3 - An anniversary clock is another typical example of a torsional pendulum.

The torsional pendulum clock is an application of the torsional pendulum and was invented around 50 years after the invention of the torsional pendulum. Such a clock is also called an "anniversary clock" or "400-day clock" because it can last a year on a single winding. This is possible because they are much more efficient than traditional regular pendulum clocks: there is less mechanical friction at the pivot but also less air resistance at the bob. The flip side of these clocks is that the string's torsion constant is very dependent on environmental factors such as temperature, so their timekeeping is less accurate than traditional regular pendulum clocks.

Torsional Pendulum Example Calculation

Let's finish by looking at some examples of what calculations involving torsional pendulums can look like.