In the previous chapters we studied how bodies move without asking what causes the motion. That description is called kinematics. Now we turn to dynamics - the study of the forces that cause and change motion. The foundation of dynamics is laid by Newton's three laws of motion, published in 1687 in the Philosophiae Naturalis Principia Mathematica. These laws describe the effect of forces on the motion of bodies and are valid for all ordinary speeds encountered in everyday life.
Newton's first law defines the concept of inertia and introduces the idea of an inertial frame of reference. His second law gives the quantitative relation between force, mass, and acceleration, F = ma. His third law states that forces always occur in equal and opposite pairs. From these laws we can derive the law of conservation of momentum and understand phenomena such as friction and the dynamics of circular motion.
This chapter is central to the entire course. It explains why a book stays at rest on a table, why a rocket moves forward, why a banked road helps a car turn, and why we lurch forward when a bus stops suddenly. Understanding Newton's laws unlocks the behaviour of almost every mechanical system.
Newton's first law of motion states that an object remains at rest or in uniform motion in a straight line unless it is acted upon by an unbalanced external force. This law is also called the law of inertia. Inertia is the property of matter by virtue of which it tends to resist any change in its state of rest or of uniform motion.
There are three types of inertia: inertia of rest, inertia of motion, and inertia of direction. When a bus suddenly starts, passengers lurch backward because of the inertia of rest; when it stops suddenly, passengers lurch forward because of the inertia of motion. A rider leaning into a turn shows inertia of direction.
The first law also defines an inertial frame of reference - a frame in which Newton's first law holds. Frames moving uniformly with respect to an inertial frame are also inertial; frames that are accelerating are called non-inertial frames. In a non-inertial frame, fictitious forces such as the centrifugal force appear.
The momentum of a body is the product of its mass and velocity:
p = m * v
Momentum is a vector quantity with SI unit kg m/s. Newton's second law states that the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force. If the mass is constant, this gives the famous relation:
F = m * a
where a is the acceleration produced. Since a = dv/dt, we can also write:
F = d(mv)/dt = dp/dt
The SI unit of force is the newton (N), where 1 N = 1 kg m/s^2. A force of one newton produces an acceleration of 1 m/s^2 on a mass of 1 kg. The second law is the fundamental law of dynamics and is used to measure force quantitatively.
Impulse is defined as the product of the average force and the time interval over which it acts. Since force is the rate of change of momentum, the impulse equals the change in momentum:
J = F * delta t = delta p
Impulse is a vector quantity with SI unit N s, which is the same as kg m/s. When a cricket ball is caught, the hands move backward with the ball to increase the time of contact, thereby reducing the average force. Similarly, spring-loaded bumpers and airbags increase the impact time and reduce the force on the occupants. The impulse is the same in all cases; only the average force changes.
Newton's third law states that to every action there is an equal and opposite reaction. If body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These two forces act on different bodies, so they never cancel each other. The action and reaction are equal in magnitude and opposite in direction, but they act on different objects.
The third law explains how a rocket works: the rocket pushes hot gases downward and the gases push the rocket upward with an equal force. It also explains recoil of a gun, walking, swimming, and rowing a boat. When we walk, we push the ground backward and the ground pushes us forward.
The law of conservation of momentum states that if the net external force acting on a system is zero, the total momentum of the system remains constant. This follows directly from Newton's second law: if F = dp/dt = 0, then p is constant.
Consider two bodies A and B interacting with each other with no external force. From Newton's third law, the force on A by B equals the negative of the force on B by A, so the changes in momentum are equal and opposite. Hence the total momentum is conserved:
m1u1 + m2u2 = m1v1 + m2v2
This law is used to analyse collisions, explosions, rocket propulsion, and recoil problems. In a rocket, as fuel is ejected backward with momentum, the rocket gains forward momentum, keeping the total momentum of the rocket-fuel system constant.
Friction is the force that opposes the relative motion between two surfaces in contact. It is a manifestation of the electromagnetic force between atoms of the surfaces. When a body just begins to move, the opposing force is the limiting friction. Static friction acts on a body at rest and can have any value up to the limiting value, while kinetic friction acts on a body in motion.
The maximum static friction (limiting friction) is proportional to the normal reaction N:
f_s = mu_s * N
and the kinetic friction is:
f_k = mu_k * N
where mu_s and mu_k are the coefficients of static and kinetic friction, respectively. Generally mu_s is greater than mu_k, which is why it is harder to start a body moving than to keep it moving. Friction can be reduced by lubrication, by rolling instead of sliding, and by polishing surfaces.
For a body of mass m moving in a circle of radius r with speed v, the centripetal acceleration is v^2/r directed toward the centre. By Newton's second law, the force causing this acceleration, called the centripetal force, is:
F_c = m * v^2 / r = m * omega^2 * r
The centripetal force is directed toward the centre of the circle. It is provided by different agencies in different situations: tension in a string, friction between tyres and road, or gravitation for a satellite.
For a vehicle taking a turn on a level road, the centripetal force is provided by friction between the tyres and the road. The maximum safe speed is v_max = sqrt(mu_s * r * g). On a banked road, the road is inclined to the horizontal, so the horizontal component of the normal reaction helps provide the centripetal force, allowing higher safe speeds.
| Law | Statement | Formula |
|---|---|---|
| First law | Body remains at rest or uniform motion unless acted on by a force | Defines inertia, inertial frame |
| Second law | Rate of change of momentum is proportional to force | F = ma = dp/dt |
| Third law | Every action has equal and opposite reaction | Forces act on different bodies |
| Quantity | Formula | SI Unit |
|---|---|---|
| Momentum | p = m v | kg m/s |
| Impulse | J = F delta t = delta p | N s |
| Static friction | f_s = mu_s N | N |
| Kinetic friction | f_k = mu_k N | N |
| Centripetal force | F_c = m v^2 / r | N |
In this chapter we studied the causes of motion through Newton's three laws. The first law introduced inertia and inertial frames, the second law gave the fundamental relation F = ma and led to the concept of momentum and impulse, and the third law explained how forces occur in equal and opposite pairs acting on different bodies. From these laws we derived the conservation of momentum, which governs collisions and rocket motion. We analysed friction, both static and kinetic, and the dynamics of circular motion with centripetal force. Newton's laws are the cornerstone of mechanics, and they will be applied repeatedly in the study of work, energy, and power in the next chapter.