Work and Energy

In everyday language 'work' means any activity that tires us — reading, thinking, running. In science, work has a strict meaning: work is done on an object only when a force acting on it produces a displacement in the object. Two conditions must both be met — a force must act on the object, and the object must be displaced. If either is missing, no work is done in the scientific sense. A student pushing hard against a wall gets tired, but since the wall does not move, the work done on the wall is zero. Similarly, a coolie carrying a load on his head and walking on level ground does no work against gravity, because the displacement (horizontal) is perpendicular to the force he is applying to support the load (vertical, upward); the work done against gravity in that horizontal motion is zero even though he feels tired.

Work is a scalar quantity — it has magnitude only, no direction. Whenever you see a force causing a displacement in its own direction, work is being done, and its amount depends on both how big the force is and how far the object moves.

Consider a constant force F acting on an object, displacing it through a distance s in the direction of the force. Scientific work done by the force is defined as the product of the force and the displacement in the direction of the force.

W = F × s

Work has only magnitude and no direction. The SI unit of work is the joule (J). One joule of work is done when a force of 1 newton displaces an object by 1 metre in the direction of the force: 1 J = 1 N × 1 m = 1 N m.

Work can be positive, negative or zero:

• Positive work: the force and displacement are in the same direction, for example lifting a bucket upward using an upward pull — cosθ = cos 0° = 1, so W = Fs.
• Negative work: the force acts opposite to the direction of displacement, for example the frictional force acting backward on a moving object, or gravity doing negative work on an object thrown upward — here θ = 180°, cosθ = −1, so W = −Fs.
• Zero work: this happens when there is no displacement (s = 0, as when pushing a wall) or when the force is perpendicular to the displacement (θ = 90°, cosθ = 0, as in a satellite moving in a circular orbit where gravity acts as the centripetal force at right angles to the velocity).

In general, W = Fs cosθ, but NCERT Class 9 mainly uses the case where force and displacement are along the same line.

An object that has the capacity to do work is said to possess energy. Energy is measured by the amount of work it can do, so energy is expressed in the same unit as work — the joule (J). A bigger unit sometimes used is the kilojoule: 1 kJ = 1000 J. When an object having energy does work on another object, some of its energy is transferred to that object, so energy can move from one body to another, and it can also change from one form to another. We meet these ideas every day: a stretched bow can shoot an arrow, moving water can turn a turbine, and the food we eat gives us the energy to work, run and think.

Energy exists in many forms in nature — mechanical energy (kinetic and potential), heat energy, chemical energy, electrical energy, light energy, sound energy and nuclear energy. Chemical energy is stored in fuels and food and is released during a chemical reaction, such as burning wood or digesting food. Heat energy makes things hot; electrical energy runs our appliances; light energy allows us to see; nuclear energy is released when the nucleus of an atom is split or fused. Mechanical energy is the energy an object possesses on account of its motion or its position, and it has two parts: kinetic energy (energy of motion) and potential energy (energy of position or configuration).

Work and energy are closely related: to do work, an object must use up some of its energy, and conversely, work done on an object increases its energy. The energy possessed by an object is thus measured in terms of its capacity to do work, and both are measured in the same unit, the joule. This link is the bridge NCERT uses before deriving the two main types of mechanical energy — kinetic energy, gained through motion, and potential energy, stored because of position or shape.

Kinetic energy is the energy possessed by an object because of its motion. A moving object can do work — a hammer in motion drives a nail, a moving cricket ball can bowl over the wickets, blowing wind can move the blades of a windmill. The faster an object moves, the more work it can do, and hence the more kinetic energy it has. An object at rest has zero kinetic energy.

Derivation: Consider an object of mass m moving with velocity u. A constant force F acts on it, bringing it to velocity v after a displacement s; let this be the only work done, so it all converts to kinetic energy.

Work done, W = F × s ... (i)

From the equation of motion, v² − u² = 2as, which gives s = (v² − u²)/2a ... (ii)

Substituting F = ma and (ii) into (i):

W = ma × (v² − u²)/2a = ½m(v² − u²)

If the object starts from rest, u = 0, so W = ½mv². This work done is equal to the change in kinetic energy of the object, so the kinetic energy of an object moving with velocity v is E_k = ½mv².

Potential energy is the energy possessed by an object because of its position or its state of configuration (shape). A stretched or compressed spring, a stretched bow, or water stored at a height all have potential energy because of their position or configuration — this energy is stored and can do work once released. For example, a stretched rubber band or bow possesses potential energy because of a change in shape and can do work when released; this is elastic potential energy. When we lift an object above the ground, we do work against gravity, and this work gets stored in the object as gravitational potential energy.

Derivation: Consider an object of mass m raised through a height h. A minimum force mg is needed to raise it without acceleration, so the work done is:

W = F × s = mg × h

This work gets stored in the object as gravitational potential energy, so the potential energy of an object at height h above the ground is E_p = mgh. This energy depends only on its position relative to a reference level, not on the path taken to reach that position.

Energy can be transformed, or converted, from one form to another. A battery converts chemical energy into electrical energy; an electric heater converts electrical energy into heat energy; a bulb converts electrical energy into light energy; a microphone converts sound energy into electrical energy and a loudspeaker does the reverse; a green plant converts light energy into chemical energy (potential energy) by photosynthesis; and burning fuel converts chemical energy into heat and light energy. These transformations are what make machines and living organisms useful — we choose the device according to the transformation we want.

The law of conservation of energy states that energy can only be transformed from one form to another; it can neither be created nor destroyed. Whenever energy seems to be 'lost' in a process, it is actually converting into other, often less useful, forms such as heat or sound due to friction and air resistance.

Illustration with a freely falling body: Consider an object of mass m at a height h above the ground, initially at rest. At this point its potential energy is E_p = mgh and its kinetic energy is zero, since v = 0, so the total mechanical energy is mgh. As the object falls, its height decreases and so does its potential energy, while its velocity, and hence its kinetic energy, increases. At any point during the fall, at a distance d fallen, the sum of the kinetic energy and the potential energy remains equal to mgh (ignoring air resistance). Just before hitting the ground, all the potential energy has converted into kinetic energy, so ½mv² = mgh at the ground, and the total mechanical energy stays constant throughout the fall — only the form changes from potential to kinetic.

Power measures how fast work is done, or how fast energy is transferred or transformed. Two people may do the same amount of work, but the one who finishes it in less time has more power. Power is defined as the rate of doing work, that is, work done divided by time taken.

P = W/t

The SI unit of power is the watt (W), named after James Watt. 1 watt is the power of an agent that does work at the rate of 1 joule per second: 1 W = 1 J/s. Bigger units are the kilowatt (1 kW = 1000 W) and horsepower (1 hp = 746 W, used mainly for engines and motors, though NCERT's SI focus is on the watt and kilowatt). When the rate of doing work is not uniform, we define average power as the total work done (or total energy consumed) divided by the total time taken.

The joule is too small to conveniently measure the large amounts of energy used by households and industries for running appliances over long periods, so a bigger unit, the kilowatt hour (kWh), is used commercially and is commonly called a 'unit' of electricity on our bills. One kilowatt hour is the energy used in one hour at 1000 joules per second (1 kW).

1 kWh = 1 kW × 1 h = 1000 W × 3600 s = 3.6 × 10⁶ J

So 1 unit of electrical energy is 3.6 × 10⁶ J. To find monthly consumption, find the power rating in kW and multiply by hours used per day and number of days — this is what electricity meters record and bills charge for.