Step from a small rowing boat onto a ghat on the Ganga. As you push off with your feet, the boat slides away behind you. Step off a large ferry in the same way and the ferry hardly moves. You pushed only on the boat, yet you moved forward. Something pushed you.
Forces come in pairs
Push a wall with your hand. You feel the wall push back on your hand. Pull a friend on skates by a rope and you are pulled towards your friend. A force is always an interaction between two bodies. Whenever body A pushes or pulls body B, body B pushes or pulls body A. The two forces of one interaction form an action-reaction pair.
When one body exerts a force on a second body, the second body exerts a force on the first that is equal in magnitude and opposite in direction.
The names "action" and "reaction" do not mean that one force comes first. The two forces appear at the same instant and vanish at the same instant. Either one may be called the action. What matters are four facts. The two forces are equal in size. They are opposite in direction. They are of the same kind; a contact push is paired with a contact push, a gravitational pull with a gravitational pull. And they act on two different bodies.
Why a pair never cancels
Two forces cancel only when they act on the same body. The two forces of a third-law pair act on different bodies, so they never cancel. Each appears in a different free-body diagram. This is how a horse can pull a cart forward. The cart pulls the horse back with an equal force, but that backward pull acts on the horse, not on the cart. Whether the cart moves depends only on the forces on the cart.
A fly hits the windscreen of a moving truck. Compare the force of the fly on the truck with the force of the truck on the fly.
Weight and normal force are not a pair
A book rests on a table. Its weight \(W\) acts downward and the normal force \(N\) of the table acts upward. They are equal in size and opposite in direction. Many students call them an action-reaction pair. They are not. Both act on the same body, the book. They are of different kinds, one gravitational and one a contact push. And they are equal only because the book is at rest; the next two chapters show them unequal.
To find the partner of a force, swap the agent and the receiver. The weight is "the Earth pulls the book". Its partner is "the book pulls the Earth", a gravitational pull on the Earth, upward. The normal force is "the table pushes the book". Its partner is "the book pushes the table", a contact push on the table, downward. Step through Fig 5.1 to see all four arrows.
Read each force as a sentence: "A pushes B". Its partner is the same sentence read backward: "B pushes A". Two forces whose sentences do not reverse into each other, like "the Earth pulls the book" and "the table pushes the book", are never a pair, however equal they look.
A coconut hangs on a tree. Which force is the third-law partner of the coconut's weight?
A boy stands on the floor of a room. Which of these are third-law pairs? Choose all that apply.
Paying off the boat
Now the boat. Your feet push the boat backward. By the third law the boat pushes your feet forward with an equal force. That forward push acts on you, so you move forward. The backward push acts on the boat, so the boat moves back. The forces are equal, but the accelerations are not: the ferry's huge mass gives it a tiny acceleration, while the small boat moves back briskly.
The plan
Every problem in the chapters that follow is solved with the same four steps. First, isolate the body whose motion you want. Second, draw its free-body diagram. Third, choose axes, one of them along the acceleration if you know its direction. Fourth, write \(\sum F = ma\) separately along each axis. Along an axis in which the body does not move, the acceleration is zero.
Isolate the body. Draw its free-body diagram. Choose two perpendicular axes, one along the acceleration. Write \(\sum F_x = m a_x\) and \(\sum F_y = m a_y\), and solve the two equations together.
A block of mass \(2\,\text{kg}\) lies on a smooth horizontal floor. A rope pulls it with a tension of \(10\,\text{N}\) at \(37°\) above the horizontal. Take \(g = 10\,\text{m/s}^2\), \(\sin 37° = 0.6\), \(\cos 37° = 0.8\). Find the acceleration of the block and the normal force of the floor.
Isolate the block. Its free-body diagram has three forces: weight \(20\,\text{N}\) down, normal force \(N\) up, tension \(10\,\text{N}\) at \(37°\). Take \(x\) along the floor, the direction of the acceleration, and \(y\) upward.
Along \(x\): \(T\cos 37° = m a\). Along \(y\): \(N + T\sin 37° - mg = 0\).
Along \(x\): \(10 \times 0.8 = 2a\), so \(a = 4\,\text{m/s}^2\). Along \(y\): \(N + 6 - 20 = 0\), so \(N = 14\,\text{N}\).
The block accelerates at \(4\,\text{m/s}^2\) along the floor. The floor pushes up with only \(14\,\text{N}\), less than the weight, because the rope lifts part of the block's weight.
A box of mass \(5\,\text{kg}\) rests on a floor. A girl pushes straight down on its top with a force of \(20\,\text{N}\). Take \(g = 10\,\text{m/s}^2\). What is the normal force of the floor on the box, in newtons?
\[ \text{Forces on the block: } mg \text{ down, } N \text{ up, } F \text{ horizontal} \]
Why. step 1 and 2 of the plan: isolate the block on a smooth floor, pushed by a horizontal force F, and draw its three forces.
\[ \sum F_y = N - mg = m a_y = 0 \]
Why. the block does not leave the floor, so the vertical acceleration is zero, and N = mg.
?
Your turn. which equation is it along the floor?
\[ a = \dfrac{F}{m} \]
Read it. divide by the mass. The acceleration is along the push.
In 1920 a New York newspaper mocked the rocket pioneer Robert Goddard. A rocket could not work in empty space, it said, because there would be nothing for the exhaust to push against. The editors had forgotten the third law. The rocket pushes the exhaust gas backward, and the gas pushes the rocket forward; no air is needed. In July 1969, as Apollo 11 flew to the Moon, the same newspaper printed a short correction.
When you walk, your foot pushes the ground backward. By the third law the ground pushes your foot forward, and that forward push is what moves you. On an icy or oily floor the foot cannot push backward on the ground, so the ground cannot push you forward, and you slip. A swimmer pushes water back; a bird pushes air down and back; a jet pushes gas back. Each is moved by the partner force.
- Calling weight and the normal force an action-reaction pair.Both act on the same body and are of different kinds. The partner of the weight is the body's pull on the Earth.
- Cancelling the two forces of a pair.A pair acts on two different bodies, so the two forces never appear in the same free-body diagram.
- Thinking the larger body exerts the larger force in a collision.The forces are equal. The accelerations differ, because the masses differ.
- Assuming the normal force always equals mg.Write the y equation. A slanting pull or an extra push changes N.