Work and Energy — Maharashtra (SSC) Class 9 Science Solutions (Free)

In physics, work is done when a force causes a displacement of an object, and is calculated as: W = F × s × cos θ where F is the applied force, s is the displacement, and θ is the angle between the direction of the force and the direction of displacement. SI unit: the joule (J), equal to one newton of force acting over one metre of displacement in the direction of that force. Work is zero in three distinct situations: 1. When the force itself is zero (nothing is pushing or pulling the object). 2. When the displacement is zero (the object doesn't actually move, no matter how much force is applied — like pushing against an immovable wall). 3. When the force is exactly PERPENDICULAR to the displacement (θ = 90°, so cos 90° = 0) — this is a common source of confusion. Example: when you carry a book horizontally at a constant height, gravity acts straight downward while your walking motion is horizontal, so the angle between gravity's force and your displacement is 90°, meaning gravity does ZERO work on the book during that horizontal walk, even though gravity is clearly still acting on it.

Given: F = 50 N, s = 10 m, θ = 30° Using W = F × s × cos θ: W = 50 × 10 × cos(30°) cos(30°) = √3/2 ≈ 0.866 W = 500 × 0.866 = 433 J The work done is 433 joules. Note why cos θ is needed at all: only the COMPONENT of the force that acts along the direction of displacement contributes to work. Here, only the horizontal component of the 50 N force (50 cos 30°) actually does work moving the body horizontally — the vertical component of the force contributes nothing to horizontal displacement.

Kinetic energy (KE) is the energy an object possesses because of its motion, given by: KE = ½mv² where m is mass and v is velocity. Example: a ball rolling across the floor possesses kinetic energy purely because it is moving — the faster it rolls, the more kinetic energy it has, and since v is squared in the formula, doubling the speed actually QUADRUPLES the kinetic energy. Potential energy (PE), specifically gravitational potential energy, is the energy an object possesses because of its position or height above a reference point, given by: PE = mgh where m is mass, g is the acceleration due to gravity, and h is height. Example: a book resting on a high shelf has gravitational potential energy relative to the floor — it has the POTENTIAL to do work (or gain kinetic energy) if it were to fall. Both forms of energy are measured in joules, and both depend directly on the object's mass — a heavier object at the same speed or height always has more energy than a lighter one.

Given: m = 5 kg, h = 4 m, g = 10 m/s² Using PE = mgh: PE = 5 × 10 × 4 = 200 J The gravitational potential energy is 200 joules — this represents the amount of kinetic energy the object would gain by the time it fell back down to its original height, if released and allowed to fall freely with no air resistance.

The Law of Conservation of Mechanical Energy states that, in the absence of friction or air resistance, the TOTAL mechanical energy (the sum of kinetic energy and potential energy) of a system remains constant — energy is not created or destroyed, only converted between these two forms. Example — a ball dropped from height h: At the top (just before release): the ball is momentarily at rest (v = 0), so all of its mechanical energy is potential energy: E = mgh, KE = 0. At the bottom (just before impact): the ball has fallen the full height h, so it has zero height remaining and therefore zero potential energy relative to the ground, but has reached its maximum speed: E = ½mv², PE = 0. If no energy is lost to air resistance, the total energy at the top exactly equals the total energy at the bottom: mgh = ½mv² — energy hasn't disappeared, it has simply CONVERTED entirely from potential form to kinetic form during the fall. At any point partway down, the ball has SOME of each — as height decreases, potential energy continuously converts into kinetic energy, while the sum of the two stays constant throughout.

Power is the RATE at which work is done — how quickly energy is transferred or converted, not just how much work is done in total. P = W/t where W is work done and t is the time taken. SI unit: the watt (W), equal to one joule of work done per second. Given: W = 500 J, t = 10 s P = 500/10 = 50 W The power is 50 watts. Two machines can do the exact same total amount of work, but the one that does it in less time has a HIGHER power rating — power measures speed of energy transfer, work alone does not.

Given: m = 2 kg, v = 6 m/s Using KE = ½mv²: KE = ½ × 2 × 6² KE = ½ × 2 × 36 KE = 36 J The kinetic energy is 36 joules.

When a force acts on an object moving at a constant velocity in the same direction as the force, power can be calculated directly as: P = F × v (This follows directly from P = W/t and W = F × s, since s/t = v.) Given: F = 20 N, v = 3 m/s P = 20 × 3 = 60 W The power delivered is 60 watts. This formula (P = Fv) is especially useful for continuously running machinery — like a car engine or a conveyor belt motor — where force and speed, rather than a one-off work and time value, are the quantities most naturally measured.

Step 1: Find the total mechanical energy at the start (height = 20 m, at rest, so all potential energy) E_total = PE = mgh = 1 × 10 × 20 = 200 J Step 2: Find the potential energy remaining at height = 10 m PE(at 10m) = mgh = 1 × 10 × 10 = 100 J Step 3: Apply conservation of mechanical energy — total energy stays constant throughout the fall E_total = KE + PE 200 = KE + 100 KE = 200 − 100 = 100 J At a height of 10 m — exactly halfway down — the ball's kinetic energy (100 J) exactly equals its remaining potential energy (100 J). This is a general result for a ball falling from rest: at the halfway point of its fall, kinetic and potential energy are always equal, each holding exactly half of the total mechanical energy.