Stand on any railway platform in India and look up. A lamp hangs from the roof by a chain. The lamp does not move all day. Yet the Earth pulls the lamp down every second, and the chain pulls the lamp up every second. Two forces act, and nothing happens. The question of this lesson is simple: when do forces produce nothing?
Forces that cancel
The lamp does not move because the two forces on the lamp cancel. The Earth pulls the lamp down with its weight \(W\). The chain pulls the lamp up with a tension \(T\). The two forces are equal in size and opposite in direction, so their vector sum is zero. The sum of all the forces on a body is called the net force. When the net force is zero, the forces produce no change in the motion of the body.
A body is in equilibrium when the net force on the body is zero. For a body at rest, equilibrium means that the body stays at rest. This state is called static equilibrium, and it is the only kind of equilibrium in this chapter.
One vector equation, two number equations
A vector is zero only when every component of the vector is zero. So the single vector equation \(\sum \vec F = 0\) becomes two ordinary equations, one for each axis. The forces along \(x\) must cancel among themselves. The forces along \(y\) must also cancel among themselves. Two equations can find two unknowns, for example two tensions, or one tension and one normal force.
What the condition does not say
The condition \(\sum \vec F = 0\) is short, and students read too much into it. Four statements are often believed and are false.
- It does not say that no force acts. The lamp has two forces on it, and a book on a table has two.
- It does not say that the forces are small. A bridge cable may carry a tension of a million newtons and still be in equilibrium.
- It does not say that a body at rest for an instant is in equilibrium. A ball thrown up stops at the top for an instant. But the only force on it is its weight, so the net force is not zero.
- It does not say that the body cannot turn. Two equal and opposite forces along different lines can rotate a body. In this chapter every body is a particle, so all forces act at one point and turning does not arise.
"At rest" and "in equilibrium" are not the same statement. A body in equilibrium stays at rest. A ball at the top of its flight is at rest only for an instant. It has a net force on it and does not stay at rest. Always test the forces, never only the velocity.
A body that moves with constant velocity also has zero net force on it, so it is also in equilibrium. A Metro coach cruising at a steady speed on a straight track is an example. This chapter deals only with bodies at rest; motion with forces is the subject of the next chapter.
A cricket ball is thrown straight up. At the highest point, the ball is at rest for an instant. Is the ball in equilibrium at that instant?
A signboard hangs at rest from two chains. Which statements are certainly true?
The free-body diagram, again
The condition \(\sum \vec F = 0\) needs a list of every force on one chosen body. You learned to draw that list in the last chapter, as a free-body diagram. The body is drawn alone, with one arrow for each force on it. This chapter is practice. Every problem starts with the same diagram, drawn again and again, until drawing it takes no thought at all.
The method, step by step
The same five steps work for every problem in this chapter. They take a minute to learn and a month of practice to make automatic.
- Choose one body. Draw it alone, as a small block or a dot.
- Draw the weight \(W = mg\), vertically down. The weight is the only force that acts without contact.
- Go round the edge of the body. Every contact gives one force. A surface gives a normal force perpendicular to the surface. A string gives a tension along the string, away from the body.
- Check that no arrow is a force that the body exerts on something else.
- Choose two perpendicular axes and write \(\sum F_x = 0\) and \(\sum F_y = 0\).
Imagine that the body is inside a closed bubble. Only two kinds of force can reach the body through the bubble. One is gravity, which needs no contact. The other is a push or pull from something that touches the bubble. Walk round the bubble with a finger and count the contacts. That count, plus one for the weight, is the number of arrows.
Forces head to tail
Draw the forces of a free-body diagram one after another, the tail of each arrow at the head of the one before. If the net force is zero, the last arrow ends exactly where the first arrow began. The arrows close into a figure called the force polygon. A closed polygon is the picture of \(\sum \vec F = 0\). For two forces the polygon is flat: the second arrow runs back along the first.
A physics book of mass \(1.2\,\text{kg}\) lies at rest on a level desk. Take \(g = 10\,\text{m/s}^2\). Draw the free-body diagram of the book and find the force that the desk exerts on the book.
The book is the chosen body. The weight is \(W = mg = 1.2 \times 10 = 12\,\text{N}\), vertically down.
The only contact is the desk, below the book. The desk pushes up with a normal force \(N\), perpendicular to its surface.
Take \(y\) vertically up. Then \(\sum F_y = N - W = 0\), so \(N = 12\,\text{N}\), vertically up.
The force of the book on the desk is also \(12\,\text{N}\), but it acts on the desk. It does not belong in this diagram.
A student draws the three forces on a body head to tail. The last arrow ends a little to the right of the start of the first arrow. What does the gap mean?
A lantern of mass \(1.5\,\text{kg}\) hangs at rest from a single vertical string. Take \(g = 10\,\text{m/s}^2\). Find the tension in the string, in newtons.
A ball hangs from a vertical string and just touches a smooth vertical wall. What force does the wall exert on the ball?
Explain why the forces on a body in equilibrium, drawn head to tail, must end where they began.
Drawing vectors head to tail adds them. The arrow from the start of the first vector to the end of the last is their sum. For a body in equilibrium that sum, the net force, is zero. A zero vector has no length, so the end point must coincide with the starting point, and the polygon closes.
A ceiling fan hangs at rest from its rod in a classroom. Roughly what is the pull of the rod on the fan, in newtons? Take \(g = 10\,\text{m/s}^2\).
A weighing machine at a railway station does not measure mass directly. The platform of the machine pushes up on your feet with a normal force, and the machine measures that push. When you stand still, you are in equilibrium, so the push equals your weight. The dial shows the push divided by \(g\), which is your mass. Lean on a wall while you stand on the machine, and the reading falls, because the wall now shares the work of balancing your weight.
- Calling a body in equilibrium because it is at rest for an instant.Test the forces. In equilibrium the net force is zero. A body at rest for an instant may still have a net force on it.
- Drawing the force that the body exerts on the table in the free-body diagram of the body.Ask of every arrow, "Which body does this force act on?" If the answer is not the chosen body, remove the arrow.
- Forgetting the weight because nothing seems to touch the body from below.Draw the weight first, every time. Gravity needs no contact.