Rotation
PHYSICS 1250
Angular Acceleration
Angular Velocity form [][Circular Motion]] was defined as \(\omega = \frac{\Delta \theta}{\Delta t}\) and the relationship between Angular Velocity and Linear Velocity is defined as \(v = \omega r\). However, like most speeds, Angular Velocity does not have to be constant, it can be effected by Angular Acceleration (\(\alpha\)) where \(\alpha = \frac{\Delta \omega}{\Delta t}\). Just like Angular Velocity, Angular Acceleration can be found in relation to Translational Acceleration with the equation \(a = r \alpha\).
Rotational Kinematics
The Kinematics for Rotational Motion describes the relationship between rotation angle, angular velocity, angular acceleration and time. Much like normal kinematics do for translational motion, in-fact they’re so similar they use essentially the same equations.
| Kinematics of Rotational Motion |
|---|
| \(\theta = \omega t\) |
| \(\omega = \omega_0 + \alpha t\) |
| \(\omega^2 - \omega^2_0 = 2\alpha \times \Delta \theta\) |
| \(\Delta \theta = \omega_0 t + \frac{1}{2}\alpha t^2\) |
| \(\Delta \theta = \frac{1}{2} ( \omega_0 + \omega ) t\) |
Rotational Dynamics
Forces also play a role on Angular Velocity and as such the equation \(F = ma\) can be rewritten for rotational dynamics using \(a = r \alpha\), so \(F = mr\alpha\).
Now using Torque, the turning effectiveness of a force from [][Circular Motion]], we know that torque is \(\tau = Fr\) because the force is perpendicular to \(r\). Now this also means that \(\tau = mr^2\alpha\). Here the quantity \(mr^2\) is referred to as rotational inertia or the moment of inertia.
Moment of Inertia
Moment of Inertia fulfills the same role that mass does but for angular problems.
The moment of inertia \(I\) is a sum of \(mr^2\) for all masses it is made of so \(I \sum_{i}{m_i r_i^2}\). this is because moment of inertia changes based on the shape of an object. The net torque of is defined using moment of inertia where \(\sum{\tau} = I\overrightarrow\alpha\).
Because moment of inertia changes based on the shape of the object, it is helpful to know the moment of inertia for different objects. Note that these will be provided on tests.
| Shape | Inertia |
|---|---|
| Hoop | \(MR^2\) |
| Disk | \(\frac{1}{2}MR^2\) |
| Rod about Center | \(\frac{1}{12}ML^2\) |
| Rod about End | \(\frac{1}{3}ML^2\) |
| Solid Sphere | \(\frac{2}{5}MR^2\) |
| Hollow Sphere | \(\frac{2}{3}MR^2\) |
Rotational Energy
The equation for Rotational Work is \(W_{net} = \tau \times \Delta \theta\). Using this and the work-energy theorem, Rotational Kinetic Energy (\(K_{rot}\)) can be found to be \(K_{rot} = \frac{1}{2}I\omega^2\).
An example or why objects don’t roll downhill at the same rate
Soup companies will often test their soup cans by rolling down an incline, the soups that roll too fast are too thin and the slowest ones are too thick. But why is this?
- Assuming there is no friction, the only work on the soup cans is that of gravity.
- This means that \(mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\). Where the greater the \(I\) is, the less energy can be in the translation and vise versa.
Angular Momentum
Angular Momentum (\(L\)) is defined as \(L=I\omega\), which is an analog of linear momentum. And just like linear momentum, Angular Momentum is conserved. So \(L_i + \Delta L = L_f\).