Force and Laws of Motion

A force is simply a push or a pull that one object exerts on another. Forces are all around us—when we push a door open, kick a ball, or a rope pulls a bucket up from a well, a force is at work. A force can change the speed of an object, change its direction of motion, or change its shape.

When two or more forces act on an object at the same time, we must look at their combined, or net, effect.

• Balanced forces are forces that are equal in magnitude but act in opposite directions on an object, so that their resultant (net) force is zero. Balanced forces do not change the state of rest or of uniform motion of an object, although they can change its shape.
• Unbalanced forces are forces whose resultant is not zero. An unbalanced force can start motion in a stationary object, speed it up, slow it down, stop it, or change its direction.

Worked example: Two children pull a toy cart from opposite ends. Child A pulls with a force of 15 N towards the east and child B pulls with a force of 15 N towards the west at the same time.

Net force = 15 N (east) − 15 N (west, taken as negative) = 0 N.

Since the net force is zero, the forces are balanced and the cart remains at rest (though the rope may stretch slightly, showing that balanced forces can still change shape).

Now suppose child A pulls with 20 N (east) while child B still pulls with 15 N (west). The net force = 20 N − 15 N = 5 N towards the east. This unbalanced force of 5 N will make the cart accelerate towards the east.

Newton's first law of motion states: An object remains in a state of rest or of uniform motion in a straight line unless it is acted upon by an unbalanced (external) force.

This law tells us that objects do not change their state of motion on their own; a force is needed to make them start moving, stop moving, speed up, slow down, or change direction. Because of this, Newton's first law is also called the law of inertia.

Inertia is the natural tendency of an object to resist any change in its state of rest or of uniform motion. The inertia of an object is measured by its mass—the more massive an object, the greater its inertia, and the harder it is to change its state of motion. This is why it is easier to push an empty trolley than a fully loaded one.

There are three common types of inertia observed in everyday life:

• Inertia of rest: the tendency of a body to continue in its state of rest. Example: when a bus starts moving suddenly, standing passengers tend to fall backward because their feet (in contact with the bus floor) start moving with the bus, but the upper part of their body tends to remain at rest.
• Inertia of motion: the tendency of a moving body to continue moving. Example: when a running bus stops suddenly, passengers tend to fall forward because the lower part of their body stops with the bus, but the upper part continues moving due to inertia.
• Inertia of direction: the tendency of a body to resist a change in the direction of motion. Example: mud flies off a spinning bicycle wheel tangentially because the mud particles tend to continue moving in a straight line.

Worked example: A coin is placed on a piece of cardboard resting on the mouth of a glass. When the cardboard is flicked away sharply, the coin drops straight into the glass instead of flying off with the cardboard. This happens because the coin, due to its inertia of rest, tends to remain at its original position while the cardboard is removed quickly, and gravity then pulls the coin straight down into the glass.

The momentum of a moving object is defined as the product of its mass and its velocity. It is a vector quantity, so it has both magnitude and direction, and its SI unit is kilogram metre per second (kg m/s or kg m s⁻¹). A heavy truck moving slowly can have the same momentum as a small car moving fast.

Newton's second law of motion states: The rate of change of momentum of an object is directly proportional to the applied unbalanced force, and this change takes place in the direction of the applied force.

Consider an object of mass m moving with initial velocity u. Let a force F act on it for time t, changing its velocity to v.

Rate of change of momentum = (mv − mu)/t = m(v − u)/t

Since acceleration a = (v − u)/t, the rate of change of momentum becomes ma. By Newton's second law, F ∝ ma, and choosing units such that the constant of proportionality is 1, we get the familiar equation:

F = ma

This shows that a larger force is needed to produce a large acceleration in an object, and that for the same force, an object with a larger mass will have a smaller acceleration.

Worked example: A body of mass 5 kg, moving with a velocity of 2 m/s, is acted upon by a force for 10 s, after which its velocity becomes 12 m/s. Find the force applied.

Given: m = 5 kg, u = 2 m/s, v = 12 m/s, t = 10 s

Acceleration, a = (v − u)/t = (12 − 2)/10 = 1 m/s²

Force, F = ma = 5 kg × 1 m/s² = 5 N

So a force of 5 N acted on the body in the direction of its motion.

Newton's third law of motion states: For every action, there is an equal and opposite reaction, and the action and reaction forces always act on two different bodies, simultaneously.

This means that whenever one object exerts a force (the "action") on a second object, the second object simultaneously exerts a force of the same magnitude but in the opposite direction (the "reaction") on the first object. Action and reaction forces act on different bodies, so they never cancel each other out.

Everyday examples of Newton's third law include:

• Walking: our foot pushes backward against the ground (action), and the ground pushes our foot forward with an equal and opposite force (reaction), which propels us ahead.
• A swimmer pushes water backward with their hands and legs, and the water pushes the swimmer forward.
• A gun recoils backward when a bullet is fired forward from it.
• A rocket rises upward because it pushes hot gases downward and the gases push the rocket upward with an equal and opposite force.

Worked example: Block P of mass 2 kg pushes against block Q of mass 8 kg with a force of 16 N. By Newton's third law, block Q pushes back on block P with an equal and opposite force of 16 N. Find the acceleration of each block.

Acceleration of P, a_P = F/m_P = 16/2 = 8 m/s²

Acceleration of Q, a_Q = F/m_Q = 16/8 = 2 m/s²

Although the force on both blocks has the same magnitude (16 N), the lighter block P accelerates four times as much as the heavier block Q, since acceleration depends on mass as well as force.

The law of conservation of momentum states: In the absence of an external unbalanced force, the total momentum of a system of interacting objects remains constant (conserved). This law follows directly from Newton's second and third laws of motion.

Consider two objects, A and B, of masses m₁ and m₂, moving along the same straight line with initial velocities u₁ and u₂. They collide for a short time t and move off with velocities v₁ and v₂. During the collision, A exerts a force on B, and by Newton's third law, B exerts an equal and opposite force on A. Since both forces act for the same time t, the change in momentum of A is equal in magnitude and opposite in direction to the change in momentum of B. Therefore, the total momentum before the collision equals the total momentum after the collision:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Worked example: A bullet of mass 20 g (0.02 kg) is fired from a gun of mass 4 kg with a velocity of 50 m/s. Calculate the recoil velocity of the gun.

Given: mass of bullet, m₁ = 0.02 kg; velocity of bullet, v₁ = 50 m/s; mass of gun, m₂ = 4 kg; recoil velocity of gun, v₂ = ?

Before firing, both the gun and the bullet are at rest, so the total initial momentum = 0.

By the law of conservation of momentum: m₁v₁ + m₂v₂ = 0

(0.02 × 50) + (4 × v₂) = 0

1 + 4v₂ = 0, so v₂ = −1/4 = −0.25 m/s

The negative sign shows that the gun recoils (moves backward) with a velocity of 0.25 m/s, opposite to the direction in which the bullet is fired.