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doPhysics · Dynamics

Objects in Equilibrium

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Why this chapter
Why this chapter exists

Think of a lamp on a platform, a crate on a ramp, or a flag at the top of a mast. Each is at rest under several forces. The last chapter gave you the laws and the free-body diagram. This chapter is practice. You will draw the same diagram on tables, inclines and strings, against walls and over pulleys. You will draw it until it takes no thought. Every problem ends with one condition, that the forces add to zero, and two ways of using it.

You will learn
By the end of this chapter you will
  1. State the condition for equilibrium, write it as two component equations, and say what it does not claim.
  2. Draw the free-body diagram and the force polygon on a table, an incline and strings, against a wall and over a pulley.
  3. Choose between the method of components and the triangle of forces, and prove and use Lami's theorem.
  4. Find the normal force in any situation, and explain why it is often not mg.
  5. Predict the readings of spring balances in series, find the stretch of springs in series, and catch the three common errors in a free-body diagram.
Before you begin
Before you begin

You should know the forces of the last chapter (weight, normal force, tension and the spring force) and how to draw a free-body diagram. You should be able to split a vector into components along two perpendicular axes, and use sine and cosine for any angle. The sine rule for a triangle is used once, in Lesson 3.

Already know this?

Lessons

1
The Condition for Equilibrium
When the forces on a body add to zero, the body is in equilibrium. What that condition says, what it does not say, and the free-body diagram that lets you use it.
Sum of forces is zeroFree body diagram and force polygon
14 min0%
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2
Inclines and Strings
Two situations that come back in every chapter of mechanics. A block held on a smooth incline by a string, and a load hung from two strings. Both are solved with the same free-body diagram and two equations.
Block on a smooth inclineHanging from two strings
15 min0%
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3
Components, the Triangle and Lami's Theorem
Two ways to solve the same knot. Resolve every force into components, or draw the closed triangle of forces and use its angles. Lami's theorem is the triangle method written as one line.
Method of componentsTriangle of forcesLamis theorem
16 min0%
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4
A Push and a Pulley
A body held aside by a string and a horizontal push, and a light pulley that turns a tension round a corner. Both are three-force problems in disguise.
String and horizontal pushPulley changes direction
15 min0%
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5
The Normal Force Is Not Always mg
A surface pushes only as hard as it must. Push down, pull up, tilt the surface or add a string, and the normal force changes while the weight stays the same.
Normal force is not always mgFinding the normal force
15 min0%
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6
Springs in Series and Errors in a Diagram
A light string or spring carries one force along its whole length, so balances in series all read the same. Then the three errors that spoil most free-body diagrams, and the checks that catch them.
Strings and springs in seriesErrors in free body diagrams
14 min0%
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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.

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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.

component
The part of a vector along one chosen axis. A force in a plane has two components along two perpendicular axes.
concurrent forces
Forces whose lines of action pass through one point. All the forces on a particle are concurrent.
equilibrium
The state of a body when the net force on it is zero. A body in equilibrium that starts at rest stays at rest.
force polygon
The figure formed by drawing all the forces on a body head to tail. For a body in equilibrium the polygon is closed.
free-body diagram
A drawing of one body alone with an arrow for every force acting on it, and for nothing else.
incline
A flat surface tilted at an angle \(\theta\) to the horizontal, also called an inclined plane.
Lami's theorem
For three concurrent forces in equilibrium, each force divided by the sine of the angle between the other two is the same for all three.
light pulley
A pulley whose mass is negligible and which turns without friction. It changes the direction of a string, but not its tension.
light string
A string whose mass is negligible. Its tension is the same at every point along it.
net force
The vector sum of all the forces acting on a body, \(\sum \vec F\).
normal force
The push of a surface on a body in contact with it, perpendicular to the surface. Its size is whatever equilibrium requires, and it can never be negative.
smooth surface
A surface with no friction. It can push on a body only perpendicular to itself.
spring balance
A light spring with a scale, which reads the tension in the spring, the force pulling at either end.
spring constant
The force needed per unit extension of a spring, \(k = F/x\), in N/m. A stiff spring has a large \(k\).
springs in series
Springs joined end to end so that the same force runs through each. Their stretches add, and \(1/k = 1/k_1 + 1/k_2\).
tension
The pull of a string or rope on a body tied to it, along the string and away from the body.
triangle of forces
The force polygon of three forces in equilibrium. Its sides are the three forces and its angles come from their directions.
weight
The gravitational pull of the Earth on a body, \(W = mg\), directed vertically downward.