2. Work and Energy

1. Meaning: potential energy is the energy stored in an object because of its specific state or position, whereas kinetic energy is the energy an object has because of its motion.
2. Formula: potential energy of an object at height h is P.E. = mgh, whereas kinetic energy of an object moving with velocity v is K.E. = ½mv².
3. Depends on: potential energy depends on the mass, the height (position) or the state of the object; kinetic energy depends on the mass and the velocity of the object.
4. Object at rest: an object at rest can have potential energy, but its kinetic energy is zero.
5. Relative nature: potential energy depends on the reference level from which height is measured (for Ajay and Atul the same ball has different P.E.); kinetic energy depends on the speed of the object.
6. Examples: water in an overhead tank, a stretched bow, a compressed spring and a wound-up clock spring have potential energy. A moving car, a flying arrow, a rolling ball and flowing water have kinetic energy.

Suppose an object of mass m is initially at rest (u = 0). A force F acts on it and produces an acceleration a. After time t its velocity becomes v and its displacement is s.

Let an object of mass m be released from rest at point A, at height h above the ground. Let B be a point at distance x below A, and C the point on the ground.

From (1), (2) and (3), the total energy is the same (mgh) at every point of the fall. As the object falls, its potential energy decreases and its kinetic energy increases by exactly the same amount. On reaching the ground the potential energy is zero and the kinetic energy is mgh — equal to the potential energy it had at the start. Hence the kinetic energy of a freely falling object on reaching the ground is nothing but its initial potential energy transformed into kinetic energy.

Let F be the applied force and s the displacement, with an angle θ = 30° between them. The component of the force in the direction of the displacement is F₁ = F cos θ.

So when the displacement is at 30° to the force, the work done is W = (√3/2) Fs ≈ 0.866 Fs, i.e. about 86.6% of the work that the same force would do if the displacement were along the force. For example, if F = 10 N and s = 2 m, W = 10 × 2 × 0.866 = 17.32 J.

No. An object with zero momentum has no kinetic energy.

• Momentum p = mv. The mass m of an object is never zero, so if p = 0, the velocity v must be zero — the object is at rest.
• Kinetic energy K.E. = ½mv². If v = 0, then K.E. = ½ × m × 0 = 0.
• The two quantities are directly related: K.E. = p² / 2m. When p = 0, K.E. = 0.

So an object with zero momentum is at rest and has zero kinetic energy. It may still have potential energy, depending on its position.

• In uniform circular motion, the force that keeps the object on the circle acts along the radius, towards the centre.
• At every point, the displacement of the object is along the tangent to the circle.
• The radius and the tangent are perpendicular to each other, so the angle between the force and the displacement is θ = 90°.
• W = Fs cos 90° = Fs × 0 = 0.

Therefore the work done on an object in uniform circular motion is zero. That is also why its speed, and hence its kinetic energy, does not change. For example, the gravitational force of the earth does no work on an artificial satellite moving in a circular orbit.

• (i) transferred from one place to another  and  (iii) transformed from one type to another

When work is done, energy passes from the object doing the work to the object on which work is done, and it usually changes form — for example, the chemical energy in our muscles becomes the potential energy of a bag we lift. Energy can never be destroyed, and simply concentrating it does no work.

• (ii) work  and  (iv) energy

Energy is the capacity to do work, so both have the same unit, the joule. The unit of force is the newton and that of power is the watt.

• (ii) gravitational force  and  (iii) reaction force in vertical direction

The object does not move up or down, so the downward gravitational force (its weight) is balanced by the upward reaction of the surface. On a smooth surface there is practically no friction, and the horizontal applied force is not balanced by any force — it accelerates the object.

• (i) the rapidity with which work is done

Power is the rate of doing work: P = W / t.

• (ii) gravitational force  and  (iii) frictional force

While lifting, the object moves up but gravity acts downward (θ = 180°), so gravity does negative work. While dragging, the object moves forward but friction acts backward, so friction does negative work. The applied force does positive work, and the vertical reaction force does zero work because it is perpendicular to the displacement.

Potential energy depends on height (P.E. = mgh). When you lie on the ground, your body is at the lowest possible height, so its potential energy is the least.

During free fall, potential energy is converted into kinetic energy, but their sum stays equal to mgh at every point (law of conservation of energy).

Potential energy depends on height, not on speed. On a flat surface the height does not change, so the potential energy stays the same. (Its kinetic energy, however, becomes 4² = 16 times.)

W = Fs cos θ depends only on the applied force, the displacement and the angle between them. The initial velocity does not appear in it.

At the moment of release the balls are at rest at a height, so they have only potential energy (P.E. = mgh). Their kinetic energy is zero.

As the balls roll down, their height decreases and their speed increases. So potential energy is converted into kinetic energy.

Both balls have the same mass and start from the same height, so they have the same potential energy at the top. The length of the channel does not matter. At the bottom, the whole of this potential energy has become kinetic energy, so both balls reach the floor with the same kinetic energy and the same speed. With equal energy they roll the same distance on the floor.

At the bottom of the channels the total energy of the balls is in the form of kinetic energy. As they roll along the floor and finally stop, this kinetic energy is gradually changed into heat (and a little sound) because of friction.

The activity demonstrates the law of conservation of energy: energy can neither be created nor destroyed; it can only be converted from one form into another, and the total energy remains constant. The potential energy the balls have at the top is not lost; it is converted into an equal amount of kinetic energy at the bottom. That is why both balls, whatever the length of the channel, end up with the same energy and cover the same distance.

Answer: the pump lifts about 1224.5 kg of water every minute.

Answer: the iron consumes 18 units of electricity in April.

Answer: the ball will rebound to a height of 6 m.

Answer: the work done on the car is 131250 J.

Answer: Ravi did 3 J of work.

Some everyday examples to start with:

For each instance, note where the energy comes from, what form it changes into, and whether some of it is also wasted as heat or sound.