Welcome to Rigid Body Mechanics (HL)!
Hello future physicist! This HL topic, Rigid Body Mechanics, takes the concepts you learned in linear motion (A.1, A.2, A.3) and applies them to objects that are spinning or rotating. Don't worry if this sounds complicated—it’s mostly about learning a new set of rotational vocabulary and seeing how everything we know about force and acceleration has a perfect rotational twin!
This section is crucial because most real-world motion involves rotation, whether it’s a car wheel, a spinning galaxy, or a figure skater. Understanding this transition from linear to rotational physics is a hallmark of Higher Level study in the "Space, time and motion" section.
1. Defining the Rigid Body and Rotational Kinematics
What is a Rigid Body?
A rigid body is a physical object that does not deform. That is, the distance between any two given points on the body remains constant, regardless of external forces acting on it. While no object in the real world is perfectly rigid, this model is extremely useful for simplifying rotational calculations (like analyzing a spinning bicycle wheel or a solid disk).
1.1 Introducing Angular Kinematics (A.4.1)
When an object rotates, we can no longer describe its motion using just linear displacement (\(s\)), velocity (\(v\)), and acceleration (\(a\)). We need new, angular variables.
- Angular Displacement (\(\theta\)): How far an object has rotated.
- Measured in radians (rad). Remember, \(360^\circ = 2\pi\) rad.
- Angular Velocity (\(\omega\)): The rate of change of angular displacement. This is the rotational speed.
- Formula: \(\omega = \frac{\Delta \theta}{\Delta t}\)
- Measured in radians per second (\(\text{rad s}^{-1}\)).
- Angular Acceleration (\(\alpha\)): The rate of change of angular velocity.
- Formula: \(\alpha = \frac{\Delta \omega}{\Delta t}\)
- Measured in radians per second squared (\(\text{rad s}^{-2}\)).
Analogy: If you drive a car in a straight line, your velocity is \(v\). If you start spinning a tire, its angular velocity is \(\omega\).
1.2 Connecting Linear and Angular Variables
If a particle is moving in a circle of radius \(r\), its linear speed \(v\) is directly related to its angular speed \(\omega\):
$$\text{Linear Speed: } v = r \omega$$
Similarly, linear acceleration \(a\) is related to angular acceleration \(\alpha\):
$$\text{Linear Acceleration: } a = r \alpha$$
Important Point: These relationships only hold true if you use radians for angular measurement!
1.3 Rotational SUVAT Equations (A.4.2)
Just like linear motion, if the angular acceleration (\(\alpha\)) is constant, we can use a set of equations perfectly analogous to the linear SUVAT equations.
Memory Aid: The Rotational Dictionary
To get the rotational equations, simply replace the linear variables with their rotational counterparts:
\(s \rightarrow \theta\) (displacement)
\(v \rightarrow \omega\) (velocity)
\(a \rightarrow \alpha\) (acceleration)
\(t \rightarrow t\) (time - stays the same!)
| Linear (SUVAT) | Rotational (\(\theta \omega \alpha\)) |
|---|---|
| \(v = u + at\) | $$\omega_f = \omega_i + \alpha t$$ |
| \(s = ut + \frac{1}{2} at^2\) | $$\theta = \omega_i t + \frac{1}{2} \alpha t^2$$ |
| \(v^2 = u^2 + 2as\) | $$\omega_f^2 = \omega_i^2 + 2 \alpha \theta$$ |
| \(s = \frac{(u+v)}{2} t\) | $$\theta = \frac{(\omega_i + \omega_f)}{2} t$$ |
2. The Cause of Rotation: Torque (\(\tau\)) (A.4.3)
In linear motion, force (\(F\)) causes acceleration (\(a\)). In rotational motion, torque (\(\tau\)) causes angular acceleration (\(\alpha\)). Torque is sometimes called the "moment of force."
2.1 Defining Torque
Torque measures how effective a force is at causing an object to rotate around a specific axis.
Analogy: Opening a Door
Imagine opening a heavy door.
- If you push right next to the hinge (the axis of rotation), it's very hard to open. The distance \(r\) is small.
- If you push perpendicular to the door far from the hinge (the handle), it's easy. The distance \(r\) is large.
- If you push directly towards the hinge, it won't rotate at all. The angle is wrong.
Torque depends on three things: the magnitude of the force \(F\), the distance from the axis of rotation \(r\), and the angle \(\theta\) between \(F\) and \(r\).
$$\tau = r F \sin \theta$$
- \(r\) is the distance from the pivot point (axis of rotation).
- \(F\) is the magnitude of the applied force.
- \(\theta\) is the angle between the force vector and the radius vector.
The units of torque are newton meters (\(\text{N m}\)).
2.2 Maximizing Torque
To maximize rotation, you want \(\sin \theta = 1\), meaning \(\theta = 90^\circ\). You must apply the force perpendicular to the radius vector.
2.3 Net Torque and Equilibrium
For an object to be in rotational equilibrium (either stationary or rotating at a constant angular velocity), the net torque acting on it must be zero.
$$\tau_{\text{net}} = 0$$
3. Resistance to Rotation: Moment of Inertia (\(I\)) (A.4.3)
In linear motion, mass (\(m\)) is a measure of an object's inertia—its resistance to being accelerated. In rotational motion, we use Moment of Inertia (\(I\)).
3.1 The Concept of Moment of Inertia
Moment of inertia is the rotational equivalent of mass. It measures an object's resistance to changes in its angular velocity.
Crucial Insight: Unlike mass, the moment of inertia depends not just on the total mass, but crucially on how the mass is distributed relative to the axis of rotation.
Analogy: Spinning a Pole
Imagine you are holding a meter stick.
- If you hold it in the middle and spin it (axis through the center), it's relatively easy.
- If you hold one end and try to spin it (axis through the end), it's much harder!
The total mass is the same, but placing the mass farther away from the axis increases the moment of inertia (\(I\)).
3.2 Calculating Moment of Inertia
For a single point mass \(m\) rotating at a distance \(r\) from the axis:
$$\text{Point Mass: } I = m r^2$$
For a rigid body composed of many small masses (or particles), we sum up the \(mr^2\) for every particle:
$$\text{General (Discrete): } I = \sum m_i r_i^2$$
For continuous objects (like disks, spheres, or rods), calculus is used to find \(I\). In IB Physics, you are not required to derive these, but you must be able to use the standard formulas provided in the data booklet.
Example: A hoop (or thin ring) with mass \(M\) and radius \(R\), spinning about its center, has \(I = M R^2\). A solid disk of the same mass and radius has \(I = \frac{1}{2} M R^2\).
Did you know? Because the disk's mass is distributed closer to the axis, its moment of inertia is smaller, meaning it is easier to rotate than the hoop!
4. Rotational Dynamics and Energy
4.1 Newton’s Second Law for Rotation (A.4.3)
We can now combine torque (\(\tau\)), moment of inertia (\(I\)), and angular acceleration (\(\alpha\)) into the rotational version of \(F = ma\).
$$\text{Newton's Second Law (Linear): } F_{\text{net}} = m a$$
$$\text{Newton's Second Law (Rotational): } \tau_{\text{net}} = I \alpha$$
This equation is the foundation for solving problems involving accelerated rotation!
4.2 Rotational Kinetic Energy (\(E_k\)) (A.4.4)
A rotating body possesses kinetic energy, just like a translating body. The rotational kinetic energy depends on its rotational inertia (\(I\)) and its angular speed (\(\omega\)).
$$\text{Kinetic Energy (Linear): } E_k = \frac{1}{2} m v^2$$
$$\text{Kinetic Energy (Rotational): } E_k \text{ rotation} = \frac{1}{2} I \omega^2$$
4.3 Rolling Objects (Combined Motion)
When an object like a wheel or a ball rolls without slipping, it is undergoing two types of motion simultaneously:
- Translation (linear motion of the center of mass).
- Rotation (spinning around the center of mass).
The total kinetic energy of a rolling object is the sum of these two components:
$$E_{k, \text{total}} = E_{k, \text{linear}} + E_{k, \text{rotation}}$$
$$E_{k, \text{total}} = \frac{1}{2} M v^2 + \frac{1}{2} I \omega^2$$
Application: Rolling Down a Hill
Consider a solid sphere and a hollow hoop (both same mass \(M\) and radius \(R\)) rolling down an incline. They start with the same gravitational potential energy (\(E_p = Mgh\)).
When they reach the bottom, all that \(E_p\) has converted into \(E_{k, \text{total}}\).
$$Mgh = \frac{1}{2} M v^2 + \frac{1}{2} I \omega^2$$
Because the hoop has a higher moment of inertia (more mass far from the center) than the sphere, it converts more of the potential energy into rotational kinetic energy and less into linear kinetic energy. Therefore, the sphere will reach the bottom first because it achieves a higher linear speed \(v\).
5. Angular Momentum (\(L\)) and Conservation (A.4.5)
In linear dynamics, we learned that momentum (\(p = mv\)) is conserved if the net external force is zero. Similarly, in rotational dynamics, angular momentum (\(L\)) is conserved if the net external torque is zero.
5.1 Defining Angular Momentum
Angular momentum is the rotational equivalent of linear momentum.
$$\text{Linear Momentum: } p = m v$$
$$\text{Angular Momentum: } L = I \omega$$
Units for angular momentum are \(\text{kg m}^2 \text{s}^{-1}\).
5.2 The Law of Conservation of Angular Momentum
If the net external torque acting on a system is zero, the total angular momentum of the system remains constant.
$$\text{If } \tau_{\text{net}} = 0 \text{, then } L_{\text{initial}} = L_{\text{final}}$$
$$I_1 \omega_1 = I_2 \omega_2$$
Classic Example: The Ice Skater
A figure skater starts spinning slowly with her arms outstretched. She then pulls her arms in close to her body.
- Arms Out: Mass is far from the axis, so Moment of Inertia (\(I\)) is large. Angular velocity (\(\omega\)) is small.
- Arms In: Mass is pulled closer to the axis, so Moment of Inertia (\(I\)) decreases significantly.
Since no significant external torque acts on the skater, \(L\) must be conserved. Because \(I\) decreases, \(\omega\) must increase dramatically, causing her to spin much faster!
6. Synthesis: The Linear-Rotational Comparison (A.4.6)
The most helpful way to master rigid body mechanics is to understand the exact analogy between linear and rotational concepts. This table summarizes the core concepts for the "Space, time and motion" section.
| Linear Concept | Variable/Equation | Rotational Concept | Variable/Equation |
|---|---|---|---|
| Displacement | \(s\) (m) | Angular Displacement | \(\theta\) (rad) |
| Velocity | \(v\) (\(\text{m s}^{-1}\)) | Angular Velocity | \(\omega\) (\(\text{rad s}^{-1}\)) |
| Acceleration | \(a\) (\(\text{m s}^{-2}\)) | Angular Acceleration | \(\alpha\) (\(\text{rad s}^{-2}\)) |
| Inertia (Mass) | \(m\) (kg) | Moment of Inertia | \(I\) (\(\text{kg m}^2\)) |
| Cause of Motion | Force (\(F\)) (N) | Cause of Rotation | Torque (\(\tau\)) (\(\text{N m}\)) |
| Newton's Second Law | \(F_{\text{net}} = m a\) | Rotational Law | \(\tau_{\text{net}} = I \alpha\) |
| Kinetic Energy | \(E_k = \frac{1}{2} m v^2\) | Rotational Kinetic Energy | \(E_{k, \text{rot}} = \frac{1}{2} I \omega^2\) |
| Momentum | \(p = m v\) | Angular Momentum | \(L = I \omega\) |