doPhysics · Rotational Mechanics
Rigid Bodies and Moment of Inertia
Bodies with size and shape. The rigid body and its three kinds of motion. The moment of inertia, the rotational twin of mass, for the rod, ring, disc, cylinder and sphere. The radius of gyration, the two theorems of axes, and bodies with holes.
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Why this chapter
Why this chapter exists
So far every body has been a point. A point can move, but it cannot spin, and it cannot topple. Real bodies do both: a fan turns, a door swings, a bat tumbles through the air. This chapter builds the model that can turn, the rigid body, and sorts its motions into translation, rotation and both at once. Then it finds the quantity that plays the part of mass in rotation, the moment of inertia. Every later chapter of rotational mechanics, from torque to rolling, is built on it.
You will learn
By the end of this chapter you will
- Say what the point-mass model describes, and why spinning and toppling need a rigid body.
- Use the rule that two points of a rigid body have equal velocity components along the line joining them.
- Tell translation, rotation about a fixed axis and general motion apart, and describe general motion as translation plus rotation.
- Derive \(I = \sum mr^2\) from the kinetic energy of a spinning body, and find \(I\) for point masses about any axis.
- Find \(I = \int r^2\,dm\) for a rod, a ring, a disc, a cylinder and a sphere, and use the table of standard bodies.
- Use the radius of gyration and the parallel-axis theorem, and avoid its classic trap.
- Use the perpendicular-axis theorem for flat bodies, and find the moment of inertia of a body with a hole.
Before you begin
Before you begin
From Kinematics of Circular Motion you should know angular position, velocity and acceleration, and the link \(v = \omega r\). You should know the equations for a constant angular acceleration. From Work, Kinetic Energy and Power you should know the kinetic energy \(\tfrac12 mv^2\). You should be able to integrate a polynomial.
Lessons
1
What a Point Mass Cannot Do
The point-mass model follows only the centre of a body, so it cannot describe spinning or toppling. A rigid body keeps every distance between its particles fixed. So two of its points always have equal velocity components along the line joining them.
Point mass model describes only the centreA rigid body keeps every distance fixedEqual velocity components along the line joining two points
2
Translation, Rotation and Both
The three kinds of rigid-body motion. In translation every point moves alike, even on a curved path. In rotation about a fixed axis every point shares one \(\omega\) and one \(\alpha\). Any other motion is a translation of one point plus a rotation about it.
Translation every point moves alikeRotation about a fixed axisGeneral motion is translation plus rotation
3
Rotational Inertia
The kinetic energy of a spinning body is \(\tfrac12 I\omega^2\), and the quantity \(I = \Sigma mr^2\) that appears there is its moment of inertia. It depends on the mass, on how far the mass lies from the axis, and on the axis itself. Every distance is measured from the axis, never from the origin.
Moment of inertia from kinetic energyWhat the moment of inertia depends onPoint masses measured from the axis
4
Moment of Inertia of Continuous Bodies
For a solid body the sum \(\Sigma mr^2\) becomes the integral \(\int r^2\,dm\). The element is chosen with all its mass at one distance from the axis. This gives the rod, the ring, the disc, the cylinder, the sphere and the shell, and the table of standard bodies.
From a sum to an integralRings discs and cylindersSpheres shells and the table
5
The Radius of Gyration and the Parallel-Axis Theorem
The radius of gyration \(k\) packs the whole mass at one distance, \(I = Mk^2\). The parallel-axis theorem, \(I = I_{cm} + Md^2\), moves an axis away from the centre of mass. It shows that the centre-of-mass axis has the least \(I\). Its trap: it links an axis only to the centre-of-mass axis.
Radius of gyrationParallel axis theoremThe parallel axis trap
6
The Perpendicular-Axis Theorem and Bodies with Holes
For a flat body, I about an axis perpendicular to it is the sum of I about two perpendicular axes in its plane. This gives the diameters of a disc and a ring, the in-plane tangent and the axes of a square plate. A body with a hole is the whole body minus the plug, both about the same axis.
Perpendicular axis theoremDiameters and tangentsBodies with cavities
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- axis of rotation
- The line about which a body turns. Particles on it do not move; every other particle moves on a circle centred on it.
- cavity
- A hole or hollow cut out of a body. Its effect on the moment of inertia is found by subtracting the piece removed.
- centre of mass
- The one point of a body whose motion follows from the net force alone, as if all the mass were there. For a uniform rod, disc or sphere it is the geometric centre.
- deformable body
- A body whose particles can move closer together or further apart, such as a spring, a rubber band or a balloon.
- flywheel
- A heavy wheel with most of its mass in the rim. Fitted to a machine, its large moment of inertia keeps the turning steady.
- general motion
- Any motion of a rigid body that is neither pure translation nor pure rotation. It is a translation of one chosen point together with a rotation about an axis through that point.
- lamina
- A flat body whose thickness is negligible, such as a thin plate, a sheet or a flat ring.
- linear mass density
- The mass per unit length of a thin rod or wire, \(\lambda = M/L\) for a uniform one, in kg/m.
- mass element
- A very small piece of a body, of mass \(dm\), chosen so that all of its mass lies at one distance from the axis.
- moment of inertia
- The measure of how hard a body is to set turning or to stop about a given axis. It is \(I = \sum mr^2 = \int r^2\,dm\), with \(r\) the perpendicular distance from the axis, in kg m².
- parallel-axis theorem
- The rule \(I = I_{\text{cm}} + Md^2\). It links the moment of inertia about any axis to that about the parallel axis through the centre of mass, a distance \(d\) away.
- perpendicular-axis theorem
- The rule \(I_z = I_x + I_y\) for a lamina, with x and y in its plane, z perpendicular to it, and all three axes meeting at one point.
- point particle
- A body treated as a single point that has mass but no size. It can move, but it cannot turn or topple.
- radius of gyration
- The radius \(k\) of a ring that would hold the whole mass of a body with the same moment of inertia, \(I = Mk^2\).
- rigid body
- A body in which the distance between any two particles stays fixed, whatever forces act on it. It can move and turn but never changes shape.
- rotation about a fixed axis
- Motion in which every particle of a body moves on a circle centred on one fixed line. All particles share one angular velocity and one angular acceleration.
- surface mass density
- The mass per unit area of a thin plate or disc, \(\sigma = M/A\) for a uniform one, in kg/m².
- translation
- Motion in which every particle of a body has the same velocity at each instant. The body keeps its orientation, although its path may be curved.