8.2 Force, Gravitation, Work & Energy

Why Force and Energy Matter for RRB Group D

Railways are about heavy bodies in motion. Newton's laws explain how a locomotive pulls a train, why brakes need a long distance to stop a moving rake, and why a shunter must cushion the impact while coupling wagons. RRB Group D physics consistently features one or two questions on Newton's laws, momentum, gravitation, and the work-energy theorem.

Newton's Laws of Motion

1. First law (inertia): A body at rest stays at rest and a body in motion stays in uniform motion unless acted on by an external force.
2. Second law: The net force on a body equals the rate of change of momentum, F = dp/dt = ma (when mass is constant).
3. Third law: For every action there is an equal and opposite reaction. The forces act on different bodies, so they never cancel each other.

Momentum and Impulse

Momentum p = mv is a vector (kg·m/s). The impulse-momentum theorem says Ft = mv − mu.

Worked Example

A 1500 kg car moving at 20 m/s is brought to rest in 4 s by braking. Find the average braking force.

Impulse: Ft = mv − mu = 1500 × 0 − 1500 × 20 = −30,000 kg·m/s.

So F = −30,000 / 4 = −7,500 N (the negative sign indicates the force opposes motion).

Newton's Second Law: F = ma

The SI unit of force is the newton (N): 1 N is the force that gives a 1 kg mass an acceleration of 1 m/s².

Worked Example

A 2 kg block is pulled along a frictionless surface by a 10 N force. Find the acceleration.

a = F / m = 10 / 2 = 5 m/s².

Universal Law of Gravitation

Every particle attracts every other particle with a force along the line joining them, given by:

F = G · m₁m₂ / r²

where G = 6.67 × 10⁻¹¹ N·m²/kg² (universal gravitational constant), m₁ and m₂ are the masses (kg), and r is the distance between their centres (m).

Acceleration Due to Gravity (g)

For a body of mass m near Earth's surface (mass M, radius R):

g = GM / R² ≈ 9.8 m/s².

Mass vs Weight

• Mass (kg): amount of matter in a body; constant everywhere; scalar.
• Weight (N): gravitational force on a body, W = mg; varies with g; vector.

A 10 kg object weighs 98 N on Earth but only 16 N on the Moon — its mass remains 10 kg in both places.

Free Fall

When a body falls under gravity alone (no air resistance), all objects accelerate at g regardless of mass — Galileo's principle. The equations of motion apply with a = g.

Work, Energy and Power

Work (W)

Work is done when a force moves a body through a distance in the direction of the force.

W = Fs cos θ

SI unit: joule (J), where 1 J = 1 N·m. If the force is perpendicular to the displacement (θ = 90°), work done is zero — e.g., a porter carrying a load on his head does no work on the load while walking horizontally.

Kinetic Energy (K)

The energy a body has because of its motion.

K = ½mv²

Potential Energy (U)

The energy stored in a body due to its position or configuration.

U = mgh

Work-Energy Theorem

The net work done on a body equals the change in its kinetic energy: W_net = ΔK = ½mv² − ½mu².

Worked Example

A 0.5 kg ball is dropped from a height of 20 m. Find its speed just before hitting the ground. Take g = 10 m/s².

Using conservation of energy: mgh = ½mv² → v² = 2gh = 2 × 10 × 20 = 400 → v = 20 m/s.

Notice the mass cancels out — heavier and lighter balls hit the ground at the same speed in the absence of air resistance.

Power (P)

Power is the rate of doing work.

P = W / t

SI unit: watt (W), where 1 W = 1 J/s. 1 horsepower (hp) ≈ 746 W.

Worked Example

A 50 kg person climbs a 10 m staircase in 8 s. Find the power developed. Take g = 10 m/s².

Work done = mgh = 50 × 10 × 10 = 5,000 J. Power = 5,000 / 8 = 625 W.

Law of Conservation of Energy

Energy can neither be created nor destroyed; it can only change from one form to another. A falling stone converts potential energy into kinetic energy. A pendulum at its highest point has only potential energy; at the lowest point, only kinetic energy; in between, a mix. In the presence of friction some mechanical energy becomes heat, but the total energy remains constant.