Formula sheet: Laws of Motion
Laws of Motion14 min readLesson 1 of 7
1.1 The three laws
The first law defines the natural behaviour of a body with no net force: its velocity stays constant, which includes staying at rest. The property behind this is inertia, and mass is its measure. The second law relates the net force on a body to the rate of change of its momentum, which reduces to mass times acceleration when the mass is constant. The third law pairs every force with an equal and opposite force on a different body, so the two never cancel on the same object.
Forces are vectors, and the net force is their vector sum. A body in equilibrium has zero net force, so its acceleration is zero. Problems are solved by drawing a free body diagram for each body separately and writing the second law along convenient axes.
1.2 Impulse and momentum
Impulse is the product of force and the time over which it acts, and for a variable force it is the area under the force-time graph. The impulse-momentum theorem says that the impulse on a body equals the momentum change it produces. A brief strong push and a long gentle push can produce the same momentum change, which is why a fielder draws the hands back while catching a hard hit.
If no external force acts on a system, its total momentum stays constant. Internal forces between the parts of the system come in equal and opposite pairs, so they cancel in the total. This is the basis of recoil, collisions and rocket propulsion.
1.3 Friction
Static friction acts when the surfaces are stuck and takes whatever value is needed to prevent relative motion, up to a maximum equal to the coefficient of static friction times the normal reaction. Kinetic friction acts when the surfaces slide and is roughly independent of the sliding speed, equal to the coefficient of kinetic friction times the normal. Kinetic friction is usually smaller than the maximum static friction, which is why a heavy box is harder to start moving than to keep moving.
The angle of friction is the angle whose tangent is the friction coefficient. On an incline, the angle of repose is the steepest angle at which a block stays at rest, and its tangent equals the coefficient of static friction. These results let a simple incline experiment measure the friction coefficient.
1.4 Circular motion and the road
In uniform circular motion the net inward force is the centripetal force, with magnitude mass times speed squared over the radius. On a level road that force is supplied by friction, which limits the safe speed to the square root of the friction coefficient times the radius times g. On a banked road the normal reaction itself has an inward component, so the safe speed rises and no friction is needed at the design speed.
A body in a vertical circle feels the string tension or the track reaction change with position, and the minimum speed at the top of the path follows from the condition that the reaction just becomes zero. These results are used in the chapter exercises and in the JEE questions on this unit.
Exam
Newton's third law pairs act on two different bodies. Writing them as a single cancelled pair on one free body diagram is the commonest mistake in this chapter, and it usually removes the very force the question is about.