doPhysics · Rotational Mechanics

Dynamics of Rotation

Why this chapter

Why this chapter exists

The last two chapters built the moment of inertia and the torque. This chapter puts them together in the law that makes a body turn faster or slower, \(\tau = I\alpha\), the twin of \(F = ma\). With it you can find how fast a rope spins a wheel and how a bucket falls down a well. You can see why the two tensions on a heavy pulley differ. You can find how hard a hinge pushes on a falling rod. Then energy joins in. A turning body carries \(\tfrac12 I\omega^2\), and a torque does work and delivers power. And the tip of a falling rod beats a falling stone.

You will learn

By the end of this chapter you will

Derive \(\tau_{net} = I\alpha\) particle by particle, and explain why only external torques count.
Solve problems with wheels and blocks by one procedure: a free-body diagram per body, \(\tau = I\alpha\), \(F = ma\), and the link \(a = \alpha R\).
Find the acceleration and the two different tensions of an Atwood machine whose pulley has mass.
Find the angular acceleration of a hinged rod and the force of its hinge from \(\vec F_{ext} = M\vec a_{cm}\), and explain why \(\alpha\) follows the torque, not \(\omega\).
Use \(\tfrac12 I\omega^2\), the work of a torque \(\int\tau\,d\theta\) and its power \(\tau\omega\), and the work-energy theorem for rotation.
Solve hinged bodies and pulleys by energy, and use the dictionary between translation and rotation with its two warnings.

Before you begin

From Rigid Bodies and Moment of Inertia you should know \(I = \sum mr^2\), the table of standard bodies, the parallel-axis theorem and \(K = \tfrac12 I\omega^2\). From Kinematics of Circular Motion you should know \(\alpha\), \(a_t = r\alpha\) and the equations for constant \(\alpha\). You should know the Atwood machine, the work-energy theorem, conservation of mechanical energy, and \(\vec F_{ext} = M\vec a_{cm}\) for the centre of mass. The chapter on Torque and Equilibrium defines torque; this chapter recalls it in one line.

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Lessons

Newton's Second Law for Rotation

Newton's second law, applied to every particle of a rigid body, gives one law for the whole body turning about a fixed axis, \(\tau = I\alpha\). Internal forces cancel in pairs, so only external torques count. A steady pull on a rope gives a steady angular acceleration.

Newtons second law for rotationOnly external torques countA steady pull spins a wheel

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Pulleys with Mass

A fixed procedure for bodies that turn and bodies that slide together, tied by a string that does not slip, \(a = \alpha R\). A block falling from a heavy wheel, and the Atwood machine with a real pulley, where the two tensions must differ.

A string that does not slipBlock hanging from a wheelAtwood machine with a heavy pulley

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Hinged Bodies and Hinge Forces

A rod hinged at one end and released from the horizontal. Its angular acceleration at the start, and why its tip falls faster than \(g\). The force of the hinge from the motion of the centre of mass, and why \(\alpha\) follows the torque, not \(\omega\).

Rod released from horizontalHinge force from the centre of massAlpha follows the torque not omega

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Rotational Kinetic Energy and the Work of a Torque

A turning body carries the kinetic energy \(\tfrac12 I\omega^2\), which is ordinary kinetic energy added over its particles. A torque does work \(\tau\,d\theta\) on each small turn and delivers power \(\tau\omega\). The net work of all torques equals the change in \(\tfrac12 I\omega^2\), for a flywheel spinning up or a wheel being braked.

Rotational kinetic energyWork done by a torquePower delivered by a torqueWork energy theorem for rotation

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Energy Conservation with Rotation

The energy method for a body on a smooth hinge, which finishes the falling rod and gives its hinge force at the bottom. The tip of the rod beats a falling stone. A block and a heavy pulley solved by two routes, and the dictionary between translation and rotation with its two warnings.

Energy method for a hinged bodyThe tip beats a falling stoneBlock and pulley by two routesThe translation rotation dictionary

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

centre of mass: The point of a body that moves as if all the mass were there and all the external forces acted on it, \(\vec F_{ext} = M\vec a_{cm}\). For a uniform rod it is the middle.
constraint: A condition that ties the motions of connected bodies together. A string that does not slip on a wheel of radius \(R\) gives \(v = \omega R\) and \(a = \alpha R\).
external torque: A torque from a force applied by something outside the body. The internal forces between the parts of a rigid body give no net torque, so only external torques change its rotation.
flywheel: A heavy wheel with most of its mass in the rim, fitted to a machine. Its large moment of inertia stores kinetic energy and keeps the turning steady.
hinge: A fixed pivot about which a body can turn, such as the hinge of a door or the pin of a boom. A smooth hinge applies a force but no torque about itself.
hinge force: The force a hinge exerts on the body it holds. It has no moment arm about the hinge, so it is found from \(\vec F_{ext} = M\vec a_{cm}\), and it changes as the body moves.
moment arm: The perpendicular distance from the axis to the line along which a force acts, \(r\sin\theta\). The torque is the force times its moment arm.
revolutions per minute: A speed of turning in whole turns each minute, written rpm. Multiply by \(2\pi/60\) to get the angular velocity in rad/s.
rotational kinetic energy: The kinetic energy of a body turning about a fixed axis, \(\tfrac12 I\omega^2\). It is the ordinary \(\tfrac12 mv^2\) of its particles added up.
torque: The turning effect of a force about an axis, \(\tau = rF\sin\theta\). Here \(r\) is the distance from the axis to the point where the force acts, and \(\theta\) the angle between the radius and the force. It is measured in N m.
translation-rotation dictionary: The pairing of each quantity and law of motion along a line with its twin for rotation about a fixed axis. The quantities \(x\), \(v\), \(a\), \(m\), \(F\) pair with \(\theta\), \(\omega\), \(\alpha\), \(I\), \(\tau\).
work done by a torque: The work \(W = \int\tau\,d\theta\) done as a body turns, the area under the torque-angle graph; \(\tau\Delta\theta\) for a steady torque.