Work, Power and Energy

January 31, 2016 January 31, 2016

Work is said to have been done when a force acts on an object and the object actually moves in the direction of force. Work done is equal to the product of the force and the displacement of the object in the direction of force.

Work = Force X Displacement

SI unit of work is Joule (J) which is equivalent to SI base units 1 kg.m²/s^2.Thus, 1 Joule of work is said to have been done when a force of 1 N causes a displacement of 1 m.

Notable Examples regarding work:

No work is done by a man rowing a boat upstream but is at rest with respect to the bank. This is because when the man is rowing a boat upstream, it is at rest with respect to the bank. So, the displacement of the boat is zero. Hence, no work is done by the boat.
No work is done if I apply all the force upon a wall and is not able to move it. Similarly, I will do no work if I am a coolie and I just standing with a load on my head but not moving. {work is done in this case if I lift a luggage from ground to place it on my head}
Work can be positive, zero or negative. Negative work implies that the displacement is in opposite side of the force. Negative Work is done when brakes are applied to a moving vehicle and vehicle stops.
When a ball is projected vertically upward and it comes back due to force of gravity, work is done by both ball and gravity in opposite directions.

Work done in Circular Path

Work done depends only on the initial and final Positions and not on the actual path followed between initial and final positions. When a body moves in a circular path no work is done. This is because centripetal force acting on the body is always at right angles to the displacement of the body along the circular path. Since cos 90° = 0, so W = F cos 90° × S = 0 × S = Zero.

Similarly, when a satellite revolves around the earth in a circular orbit, the work done by force of gravity is also zero because it acts at right angles to the direction of displacement of the satellite.

Power

Power is the time rate of work done by a body. Thus if work done is divided by time taken, we get power.

Power =Work done / Time taken

The SI unit of power is Watt which is equal to 1 joule per second. 1 Horse power is equal to 746 watt. Power is a scalar quantity.

Energy

Energy of a body refers to its capacity of doing work. Energy is a scalar quantity. SI unit of Energy is erg. 1 erg = 10⁻⁷ J

There are several types of energies for example, mechanical energy, chemical energy, light energy, heat energy, sound energy, nuclear energy, electric energy etc.

Mechanical Energy

Kinetic Energy and Potential Energy are called Mechanical Energy. The sum of kinetic and potential energies at any point remains constant throughout the motion. It does not depend upon time. This is known as law of conservation of mechanical energy.

Kinetic Energy

The energy possessed by any object by virtue of its motion is called its kinetic energy.

Kinetic energy of an object is given by k = 1 / 2 mv²

where m = mass of the object, and v is its velocity.

So it’s obvious that Kinetic energy is zero in stationary objects as v=0.

The above formula shows that the Kinetic Energy is a product of half the Mass and velocity Squared. When the velocity is doubled, the Kinetic energy would go up four times. If velocity is tripled, kinetic energy would go up nine times. If velocity is increased by 1.5 times the Kinetic energy would go up by 1.5×1.5=2.25 times.

Further, since kinetic energy is a product of mass and velocity squared, a tennis ball and a football don’t have equal kinetic energy if they have equal velocities. To get equal kinetic energy, the tennis ball needs to have few times higher velocity than a football.

Energy in a running horse, Speeding car, fired bullet, oscillating pendulum, flowing water, flying bird are examples of Kinetic energy.

Potential Energy

The energy possessed by any object by virtue of its position or configuration is called its potential energy. There are three important types of potential energies viz. gravitational, elastic and electric.

If a body of mass m is raised through a height h against gravity, then it has gravitational potential energy. It would be equal to E=mgh
If a spring of spring constant k is stretched through a distance x, then elastic potential energy of the spring would be E=1/2 kx²

Examples of Potential Energy include: a stretched bow and arrow system; a wound up spring of a watch; water stored high up in reservoirs; stone lying on the top of the roof.

Work-Energy Theorem

Work energy theorem says that the work done by a force in displacing a body is equal to change in its kinetic energy. When we move an object (i.e. we do work on it), we increase its kinetic energy. When we bring a moving object to rest, we also do work on the object, but in this case we are decreasing its kinetic energy. Regardless of whether we are increasing or decreasing an object’s kinetic energy, the amount of work done is equal to the change in energy.

Mass-Energy Equivalence

Einstein showed us the way that mass can be transformed into energy. When

Δm is converted into energy, the energy produced is equal to E = Δmc^², where c is the speed of light in vacuum.

Principle of Conservation of Energy

This says that sum of all kinds of energies in an isolated system remains constant at all times.

The law of conservation of mass and energy states that the total energy (Rest mass energy + kinetic energy + potential energy) of a closed system is constant; that is, energy or mass can neither be created nor destroyed.

Principle of Conservation of Mechanical Energy

For conservative forces the sum of kinetic and potential energies of any object remains constant throughout the motion. An object may have both kinetic and potential energy at the same time but total of them would be same. For example, a flying aeroplane, an oscillating pendulum, a stone thrown upwards have both kinetic and potential energy.

A swinging pendulum has maximum kinetic energy and minimum potential energy when it is at the middle of the arc i.e. at its lowest point. However, when it is at highest point on either side, its kinetic energy is zero and all energy is potential energy. Throughout its swing, the total mechanical energy remains same.

Practical examples on Energy

Q-1: A rubber ball dropped from 24 m height and after impact it loses its kinetic energy by 25%. What is the height to which it rebounds?

Answer: In this question, all the potential energy of ball (by virtue of its being at a height of 24 m) is converted into kinetic energy when it reaches to the ground. However, the ball has lost 25% of its kinetic energy (due to inelastic collision). What remains with the ball is 75% of the kinetic energy. So it would rebound only 75% of 24 meters i.e. 18 meters.

Q-2: What kind of Energy is stored in Tides in Oceans?

Tides in the sea have stored in them combination of Hydraulic energy, Kinetic energy as well as Gravitational potential energy.

Q-3: Which of the following four objects has the least kinetic energy: an object of mass (m) moving with speed (4v), an object of mass (3m) moving with speed (2v), an object with mass (4m) moving with a speed of (v), or an object of mass (2m) moving with speed (3v)?

Answer:

Let 1/2mv² be X, so: