doPhysics · Rotational Mechanics
Torque and Equilibrium
The turning effect of a force about an axis and about a point, and the couple. The two conditions of equilibrium for a rigid body, where the weight acts, why bodies topple, and ladders and hinges.
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Why this chapter
Why this chapter exists
A force can push a body along, but it can also turn it. The same push opens a door easily at the handle and hardly at all near the hinge. This chapter measures that turning effect, the torque, and learns to add torques as carefully as forces. With torque in hand, a rigid body at rest can be understood completely. Examples are a see-saw, a plank on two supports, a leaning ladder and a hinged beam. The same ideas decide when an almirah slides and when it topples.
You will learn
By the end of this chapter you will
- Find the torque of a force about an axis in two ways, by the transverse component and by the moment arm.
- Write torque as \(\vec r \times \vec F\), find its direction by the right-hand rule, and find its component along an axis.
- Explain why a couple has the same torque about every point and cannot be balanced by one force.
- Use the two conditions of equilibrium, taking torques about the point that removes the most unknowns.
- Place the resultant of parallel forces, use the three-force rule, and explain why the weight acts at the centre of gravity.
- Decide when a body topples, and whether a pushed or tilted block slides first or topples first.
- Solve ladder problems with a smooth or rough wall, and find the force of a hinge.
Before you begin
Before you begin
From Objects in Equilibrium you should be able to draw a free-body diagram and use \(\sum\vec F = 0\) in components. From the two chapters on Friction you should know static friction and its limit \(\mu_s N\). From Rigid Bodies and Moment of Inertia you should know what a rigid body is and what rotation about a fixed axis means. You should be able to resolve a vector and multiply two vectors.
Lessons
1
The Turning Effect of a Force
Torque measures how strongly a force turns a body about an axis, \(\tau = rF\sin\theta\). Only the part of the force across the line from the axis turns. Torque is the force times its moment arm, and a spanner shows how to get the most of it.
Torque measures turning about an axisOnly the transverse component turnsMoment arm and line of actionGetting the most torque
2
Torque as a Vector and the Couple
About a point, torque is the vector \(r \times F\), with its direction given by the right-hand rule. The torque about an axis is the component of that vector along the axis. A couple has zero net force, yet the same torque \(Fd\) about every point.
Torque as a cross productTorque about a point and about an axisThe couple
3
Equilibrium of a Rigid Body
A rigid body is in equilibrium when the net force and the net torque on it are both zero. When the forces balance, torques may be taken about any point, so choose the point where the unknown forces act. The see-saw and the beam on two supports show the method.
Two conditions of equilibriumThe choice of pivot is freeBalancing a see sawA beam on two supports
4
Where the Weight Acts
Parallel forces have a single resultant whose line of action is fixed by torques; it divides the gap in the inverse ratio of the forces. Three forces in equilibrium meet at one point. The weights of all the particles of a body act as one force at its centre of gravity, which a plumb line finds.
Resultant of parallel forcesThree forces in equilibrium meet at a pointWeight acts at the centre of gravityFinding the centre of gravity by hanging
5
Toppling
A body topples when the line of its weight passes outside its base, because the normal force cannot follow it there. A push high on a block moves the normal force to the front edge. Whether a block slides first or topples first depends on its width, its height and the friction.
The weight line and the basePushing a block slide or tipSlide or topple on an incline
6
Ladders and Hinges
A ladder against a smooth wall needs friction at its foot, and the least coefficient is \(\tfrac12\cot\theta\). A steeper ladder is safer, a climber raises the friction needed, and a rough wall helps. A hinge pushes in whatever direction the balance demands, found by torques about the hinge.
Ladder against a smooth wallLeast angle and a climbing personLadder against a rough wallReaction of a hinge
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 fixed line about which a body turns. Points on it do not move; every other point moves on a circle round it.
- base of support
- The region of the floor enclosed by the points where a body touches it, such as the base of a block or the space between the feet.
- centre of gravity
- The point through which the resultant of the gravitational forces on all the particles of a body acts. In a uniform gravitational field it is the centre of mass.
- centre of mass
- The point of a body whose motion follows from the net external force alone, at \(\vec r_{\text{cm}} = \sum m_i\vec r_i / M\).
- couple
- Two equal and opposite forces acting along different parallel lines. Its net force is zero, and its torque \(Fd\) is the same about every point.
- cross product
- The product \(\vec A \times \vec B\) of two vectors: a vector of size \(AB\sin\theta\), perpendicular to both, with its direction given by the right-hand rule. Also called the vector product.
- hinge
- A joint that holds one point of a body fixed but lets the body turn about it. Its force can point in any direction, so it is found as two components.
- limiting friction
- The largest static friction a surface can give, \(\mu_s N\). A body slides once the force needed to hold it passes this limit.
- line of action
- The straight line along which a force acts, extended in both directions through its point of application.
- moment arm
- The perpendicular distance from the axis to the line of action of a force, \(d = r\sin\theta\). Also called the lever arm. The torque is \(Fd\).
- pivot
- A fixed point or edge about which a body such as a lever or a see-saw can turn. It pushes on the body with whatever force the balance needs.
- plumb line
- A string with a small heavy weight at its end. Hanging freely, it marks the vertical through its point of support.
- radial component
- The part of a force along the line from the axis to its point of application, \(F\cos\theta\). It turns nothing.
- resultant
- The single force with the same effect as several forces: their vector sum, acting along the line where its torque equals theirs.
- right-hand rule
- The rule that fixes the direction of a cross product: curl the fingers of the right hand from the first vector towards the second, and the thumb points along the product.
- rotational equilibrium
- The state of a body on which the net torque is zero, so that its angular velocity does not change. The body may be at rest or turning steadily.
- static equilibrium
- Equilibrium of a body at rest that stays at rest: the net force and the net torque on it are both zero.
- topple
- To fall by turning about an edge of the base. A body topples when its weight line passes outside its base of support.
- torque
- The turning effect of a force. About an axis it is \(\tau = rF\sin\theta\); about a point it is the vector \(\vec\tau = \vec r \times \vec F\). Measured in N m.
- translational equilibrium
- The state of a body on which the net force is zero, so that its centre of mass does not accelerate.
- transverse component
- The part of a force across the line from the axis to its point of application, \(F\sin\theta\). It does all the turning.
- uniform gravitational field
- A region in which \(\vec g\) has the same size and direction at every point. Near the Earth's surface, over the size of everyday bodies, the field is uniform to high accuracy.
- weight line
- The vertical line through the centre of gravity of a body, the line of action of its weight.