Rotational Dynamics

Rotational Dynamics Motion

We can describe rotational motion using four main physical quantities: angular velocity, angular acceleration, angular displacement, and time. While we have an intuitive notion of time, the other rotational motion variables might not sound that familiar to you, so let's start by defining each of them.

The mathematical formula corresponding to this definition is

$$\Delta \theta= \theta_f - \theta_i.$$

Angular displacement has an SI unit of radians (\(\mathrm{rad}\)).

The mathematical formula for average angular velocity is

$$\omega=\frac{\Delta{\theta}}{\Delta{t}},$$

where \( \Delta\theta \) is angular displacement and \( t \) is time. Angular velocity has a SI unit of \( \mathrm{\frac{rad}{s}} \). However, for instantaneous angular velocity we use

$$\omega=\frac{\mathrm{d}\theta}{\mathrm{d}t}.$$

We can calculate the average angular acceleration as

$$\alpha=\frac{\Delta{\omega}}{\Delta{t}},$$

and for instantaneous angular acceleration

$$ \alpha=\frac{\mathrm{d}\omega}{\mathrm{ d}t} .$$

Angular acceleration has a unit of \( \mathrm{\frac{rad}{s^2}} \).

Now besides understanding these definitions and corresponding formulas, one should also be aware of the relationship between these quantities. Angular velocity is the first derivative of angular displacement with respect to time, \( \omega=\frac{d\theta}{dt} \), while angular acceleration is the first derivative of angular velocity with respect to time, \( \alpha=\frac{d\omega}{dt} \) as well as the second derivative of angular displacement \( \alpha=\frac{d^2\theta}{dt^2}. \)

Measuring Angles with Radians

A radian is an angle measure frequently used in a circular motion.

Fig. 1 - A diagram showing the definition of a radian.

If we want to convert radians to degrees, we can use the formula,

$$\theta_\text{degrees}=\theta_\text{radians}\left(\frac{180^\circ}{\pi\;\mathrm{rad}} \right). $$

On the other hand, to convert from degrees to radians, we can apply the formula:

$$\theta_\text{radians}=\theta_\text{degrees}\left(\frac{\pi\,\mathrm{rad}}{180^\circ}\right).$$ Let's check our understanding with a quick example.

Torque and Rotational Inertia

Torque is the rotational equivalent of a force. For an object to experience angular acceleration, it has to be affected by torque.

Convention states that a counterclockwise rotation indicates positive torque, and a clockwise rotation indicates negative torque. The amount of torque applied to an object depends on the applied force but also on the perpendicular distance from where the force is applied, with respect to the axis of rotation.

For a system to be at rotational equilibrium, the sum of all torques acting on a system must equal zero.

$$\sum \tau=0$$

And we also have a quantity that plays the role of the mass in translational motion. For rotational motion, we use the moment of inertia or rotational inertia.

Rotational inertia varies depending on the shape of the object and its mass distribution with respect to the axis of rotation.

Rotational Dynamics Equations

Now let's talk about the equations of the concepts used in rotational motion and and how we can use them.

Rotational Kinematic Equations of Motion

Sometimes, it is enough to work with rotational kinematic equations when studying rotational motion.

We can use the following kinematic equations whenever we are dealing with a constant angular acceleration.

Angular velocity equation:

$$\omega=\omega_{o} + \alpha{t}.$$

Angular displacement equation:

$$\Delta{\theta} =\omega_o{t}+\frac{1}{2}{\alpha}t^2.$$

Angular velocity squared equation:

$$\omega^2={\omega_{o}}^2 +2{\alpha}\Delta{\theta}.$$

In the above equations, \(\omega\) is final angular velocity, \(\omega_0\) is the initial angular velocity, \(\alpha\) is angular acceleration, \(t\) is time, and \(\Delta \theta\) is angular displacement.

Connection Between Rotational Kinematics and Linear Kinematics

Now, as we have discussed rotational dynamics and its connection to rotational kinematics, we must be sure to also recognize and understand the relationship between rotational kinematics and linear kinematics. These relationships appear in the table below, where \( r \) is the radius of the circular motion.

To better understand these relationships, let us look at the diagram below.

Fig. 2 - A diagram illustrating the relationship between linear kinematic variables and rotational kinematic variables where \( v=\text{velocity},\) \(r=\text{radius},\) \( X=\text{linear displacement},\) \( \Delta \theta=\text{angular displacement},\) and \( \omega=\text{angular velocity.} \)

Equations for Torque

Torque can be calculated by using three different formulas.

1. Cross Product Formula
2. Magnitude Formula
3. Newtons Second Law Formula

Cross-Product Formula

The cross-product definition of torque is expressed by the equation

$$\tau=\vec{r} \times \vec{F}$$

where \( \vec{r} \) is the displacement from the axis of rotation and \( \vec{F} \) is the applied force.

The cross product is also known as the vector product. Its result is another vector quantity whose direction is perpendicular to both vector arguments and the plane defined by them. The torque vector is parallel to the axis of rotation.

Magnitude Formula

If we only care about the magnitude of the torque, \( | \vec{\tau} | = \tau, \) we can calculate it using the following formula:

$$ \tau = | \vec r \times \vec F | = r F \sin\theta.$$

where \( r \) is the magnitude of the displacement from the axis of rotation, \( F \) is the magnitude of the applied force, and \( \theta \) is the angle between them.

We can also write the above equation as

$$ \tau = Fr_\perp,$$

where \( r_\perp = r\sin\theta ,\) is the perpendicular distance from the axis of rotation to the line of action of the force, known as the lever arm.

Fig. 3 - A torque diagram showing the applied force, distance from the rotation axis, the angle they make, and the lever arm.

Newtons Second Law Form

The equation for torque can be written in the same format as that of Newton's second law, \( F=ma \).

$$\tau=I\alpha$$

Here \( I \) is the moment of inertia and \( \alpha \) is the magnitude of the angular acceleration.

Rotational Inertia Equation

The moment of inertia for a point mass at a distance of \(r\) from the axis of rotation is

$$I=mr^2.$$

This formula is important because we can use it to derive formulas for the moment of inertia of complex systems since objects can be thought of as being comprised of sets of point masses. The total rotational inertia for a collection of objects is calculated by the formula

$$I_{\mathrm{total}}= \sum I_i=\sum m_ir_{i}^2.$$

Let's try a quick example.

If our system is a finite-sized body, we can model it as being comprised of infinitely many differential masses \( \mathrm{d}m \). Each of these masses contributes to the moment of inertia of the object, but instead of adding them up, we need to integrate them to find the total moment of inertia of the object:

$$I=\int{r^2}dm,$$

where \( r \) is the perpendicular distance from the differential masses, \( dm,\) to the axis of rotation.

Equations for Other Useful Rotational Motion Quantities

We still have many other counterparts to those of translational motion. Let's talk briefly about a few of them.

Rotational Kinetic Energy

Just as velocity is used to calculate kinetic energy,

$$K=\frac{1}{2}mv^2,$$

we can quantify the rotational kinetic energy of a system knowing its angular velocity

$$K_{\text{rot}}= \frac{1}{2}I\omega^2.$$

Angular Momentum and Impulse

The mathematical formula corresponding to this definition is

$$L=I\omega,$$

where \( \omega \) is angular velocity measured in \( \mathrm{\frac{rad}{s}}, \) and \( I \) is inertia measured in \( \mathrm{kg\,m^2}. \) Angular momentum has SI units of \( \mathrm{kg\,\frac{m^2}{s}} \).

We also have a definition for the angular impulse which describes how torque, the rotational analog of force, affects a system with respect to time.

The mathematical formula corresponding to this definition is

$$\Delta \vec{J}_{rot}= \int_{t_o}^{t}\vec{\tau}(t)dt,$$

which can be simplified to

$$J=\tau\Delta{t},$$

when \( \tau \) doesn't vary with time. Since \( \tau \) is measured in \( \mathrm{N\,m} \) and \( t \) in \( \mathrm{s}, \) the angular impulse unit in the SI system is \( \mathrm{N\, \frac{m}{s}}. \)

Angular impulse and angular momentum are related by the impulse-momentum theorem:

$$ J=\Delta{L} .$$

Rotational Dynamics Experiment

To demonstrate the concept of rotational dynamics in the lab, one can complete an experiment involving a rotating arm. In this experiment, the apparatus consists of a rotating arm connected to a rotary motion sensor that measures angular velocity. On the rotating arm, two masses will be attached to slow it down by increasing its rotational inertia. The further out the masses are, the harder it is for the arm to rotate. The axis of the rotating arm will then be attached to a string which is also attached to a pulley with a hanging mass. This setup allows for a constant torque to be applied to the arm, because of tension from the string, causing it to rotate. Torque can then be measured and continuously varied using different hanging masses.

Rotational Dynamics Examples

To solve rotational dynamic problems, one can use a combination of the equations we explored before. Let us apply our new-found knowledge of rotational dynamics to the following two examples.

Now let's try another example.