A fielder catches a fast cricket ball. The palms sting, and the hands move back with the ball. A moment ago the ball was flying at thirty metres per second. Now it is at rest in the hands. Something acted on the ball to stop it. That something was a push from the fielder's hands, and it is the subject of this whole unit.
Kinematics described motion; dynamics asks why
The last unit described motion. It gave position, velocity and acceleration, and never asked what causes them. This unit asks that question. The answer is always the same: the cause of a change in motion is a force. The study of motion together with its causes is called dynamics.
A force needs two bodies
A force is a push or a pull that one body exerts on another. So every force has two parts to its name. There is the agent, the body that pushes or pulls. There is the receiver, the body that is pushed or pulled. "The hand pushes the ball" is a complete description. "The force of the ball" is not, because it names only one body.
A force is a push or a pull exerted by one body on another. A force has a magnitude and a direction and adds like a vector. Its SI unit is the newton, N.
A force has a size and a direction. Pushing a door at its handle with the same effort, first inward and then outward, gives two different results. So a force is a vector, drawn as an arrow. The arrow starts on the receiver, points the way the push or pull acts, and its length shows the size.
Contact and non-contact forces
Most forces act only while two bodies touch. A hand pushes a door, a rope pulls a bucket, a table holds up a book. These are contact forces. A contact force stops the moment the bodies separate. A few forces act across empty space. The Earth pulls a mango down from a tree although nothing touches the mango. Such a force is a non-contact force. In mechanics the only non-contact force you will need is gravity.
All the contact forces of mechanics, the push of a table, the pull of a rope, the spring force and friction, are electric forces between the atoms at the two surfaces. When a book presses on a table, the atoms of the table are squeezed a little closer, and they push back. Physicists count only four basic forces in nature: gravitational, electromagnetic, strong nuclear and weak nuclear.
A boy kicks a football. Which description of the force is complete?
Weight: the pull of the Earth
The Earth pulls every body towards its centre. This pull is the weight of the body. Near the surface of the Earth, every body falls with the same acceleration \(g\). The weight of a body of mass \(m\) is therefore \(W = mg\), directed vertically downward. A student of mass \(60\,\text{kg}\) has a weight of \(60 \times 10 = 600\,\text{N}\).
The normal force: the push of a surface
Put a book on a table. The table pushes up on the book. This push of a surface is the normal force, written \(N\). "Normal" means perpendicular. The normal force is always perpendicular to the surface of contact, and it always pushes; a surface never pulls. Its size is not fixed. The table pushes only as hard as it must, and a heavier book gets a larger push.
Tension: the pull of a string
A string tied to a body can only pull it. The pull of a string or a rope is its tension, written \(T\). Tension acts along the string and away from the body, because a string pulls towards itself. In this book a string is light and does not stretch unless the problem says otherwise. A light string has the same tension all along its length.
Applied forces, and friction by name
A hand that pushes a trolley or pulls a door exerts an applied force. It is a contact force and it acts in whatever direction the hand pushes or pulls. One more contact force acts whenever a surface is rough. The surface resists sliding along it, and this resistance is friction. Friction is parallel to the surface of contact. It has two chapters of its own later in this unit, so here we only name it.
A block rests on a ramp that is tilted at \(30°\) to the horizontal. In which direction does the ramp's normal force on the block act?
A bag of rice has a mass of \(25\,\text{kg}\). Take \(g = 10\,\text{m/s}^2\). What is its weight in newtons?
The spring force
A spring has a natural length. Stretch it and it pulls back. Compress it and it pushes back. In both cases the force is directed towards the natural length. This force of a spring on whatever holds its end is the spring force. The change in length from the natural length is the extension, written \(x\). A compression is simply a negative extension.
Watch the pull and the extension in Fig 1.2. When the extension doubles, the pull doubles. When it triples, the pull triples. The force is proportional to the extension. The constant of proportionality depends on the spring. A stiff spring needs a large force for a small extension. A soft spring stretches easily.
For a spring that is not stretched too far, the force the spring exerts is proportional to its extension and directed towards its natural length.
The number \(k\) is the spring constant. It is the force needed for an extension of one metre, so its unit is newton per metre, N/m. A spring with \(k = 200\,\text{N/m}\) pulls back with \(10\,\text{N}\) when stretched by \(5\,\text{cm}\), because \(200 \times 0.05 = 10\). The minus sign in \(\vec F = -k\vec x\) says only this: the force points opposite to the extension, back towards the natural length.
A spring of spring constant \(500\,\text{N/m}\) hangs from a hook. A bag of mass \(2\,\text{kg}\) is hung from its lower end and comes to rest. Take \(g = 10\,\text{m/s}^2\). (a) What force does the spring exert on the bag? (b) By how much has the spring stretched?
(a) The bag is at rest, so the spring must hold up its whole weight. The spring pulls up with \(mg = 2 \times 10 = 20\,\text{N}\).
(b) From \(F = kx\), the extension is \(x = F / k = 20 / 500 = 0.04\,\text{m}\), that is \(4\,\text{cm}\).
A spring stretches by \(3\,\text{cm}\) when pulled with a force of \(6\,\text{N}\). What force stretches the same spring by \(9\,\text{cm}\)?
A spring is compressed between a wall and a block. In which direction does the spring push the block?
Now return to the fielder's catch. The ball was stopped by a push of the hands: a contact force, an applied force, with the hands as agent and the ball as receiver. At the same instant the Earth was pulling the ball down with its weight \(mg\). Naming the agent and the receiver of each force told you exactly which arrows belong on the ball.
Robert Hooke, a curator of experiments at the Royal Society in London, studied springs, wires and bent wood. In 1676 he published a Latin anagram, "ceiiinosssttuv", to claim a discovery without yet revealing it. In 1678 he gave the answer: "ut tensio, sic vis", "as the extension, so the force". The same rule told him how to build a spring for a watch. It is the rule of the spring in Fig 1.2, and of every spring balance.
A hanging spring balance at a market has a spring inside it. The weight of the bag stretches the spring, and by Hooke's law the extension is proportional to the weight. So the scale can be marked in equal steps. The same spring balance would read less on the Moon, because it measures the pull on the bag, the weight, not the mass.
- Describing a force by one body only, "the force of the ball".Every force has an agent and a receiver. Say "the push of the hand on the ball".
- Drawing the normal force vertically on a slope.The normal force is perpendicular to the surface of contact. On a slope it tilts with the surface.
- Drawing tension as a push.A string can only pull. Tension points along the string, away from the body.
- Measuring the extension of a spring from the wall.Extension is measured from the natural length, not from the fixed end.