In the previous section we saw basic details about work. In this section we will see energy.
We will see energy from a scientific point of view.
1. Consider a fast moving cricket ball.
• If it hits a wicket, the stumps will be thrown away. The stumps got displaced.
• That is., the ball did some work on the stumps. For doing that work, the ball must possess some energy.
• After hitting the stumps, the energy of the ball is lost. It can no longer displace another wicket.
2. Another example: A heavy iron piece is raised to a height of about 80 cm above the ground
and let go.
• It falls upon a nail driven one fourth into a wooden piece.
• When the iron falls on the nail, the nail is driven upto three fourth into the wood
• Here the iron piece did some work on the nail. Because it displaced the nail.
• But once the nail is driven to three fourth, the energy is lost. That is., energy is all used up.
• If the iron was raised to a height more that 80 cm, it may have had sufficient energy to drive the nail completely into the wooden piece.
3. One more example: A toy car is wound and then placed on the floor.
• It starts to move. So the spring inside the car is doing some work.
• That means, the spring has some energy.
• When all that energy is used up, the toy car stops.
■ In the second example, if the iron piece was on the ground, it would not be able to drive the nail
• That is., it would not be able to do work
• That is., it would not have possessed any energy
■ In the third example, if the spring was not wound up, the toy would not have moved.
• That is., it would not be able to do work
• That is., it would not have possessed any energy
■ So we can say that, energy is the ability to do work.
■ Just as we specified a method to measure work, we need a method to measure energy also.
Let us see how energy can be measured:
• Consider an object like cricket ball, stone etc.,
♦ If it is stationary, it cannot do any work.
♦ If it is moving with some velocity, it can do some work
♦ If it is moving with a greater velocity, it can do greater work
♦ If it is moving with a lesser velocity, it can do only a lesser work
• That means, motion of a body gives it some energy.
■ The energy possessed by a body due to it's motion is called kinetic energy.
Let us summarise:
1. A moving body can do work
2. The ability to do that work comes from the energy that the body possess
3. The energy is due to the motion of the body
4. This energy possessed by a moving body is called kinetic energy
2. If it is moving with a velocity v, it has a certain amount of kinetic energy. Let us denote it as Ek
• It acquired this Ek because it acquired a velocity v
3. But the body cannot acquire a velocity v all by itself. Some one or some thing has to accelerate it from velocity zero to velocity v
4. If the body is accelerated, it is clear that some one or some thing applied a force F on the body.
• Because, Force = mass × acceleration. (Details here)
5. Now, if the body is moving with a velocity v, it is clear that, the body is being displaced
So at this stage, we know the following 4 facts:
(i) The body was at rest
(ii) It was acted upon by a force F
(iii) As a result, it was subjected to acceleration and acquired a velocity v
(iv) While increasing the velocity from zero to v, the body would have moved through a certain distance.
6. We call this distance as: 'displacement'.
• Let s be the displacement in which the velocity v was acquired
7. Then we can say this: The force F caused a displacement of s
• That means, the force F did some work (F×s) on the body. (Details in the previous section)
• This work will remain in the body moving with velocity v. And so, this work is equal to the kinetic energy.
• Thus, F×s is the kinetic energy possessed by the moving body.
♦ We can write: Ek = F×s
♦ We have to calculate this F×s. Let us try:
8. We have two terms: F and s
• F can be written as (m×a).
♦ Where m is the mass of the body and
♦ a is the acceleration which enabled the body to acquire a velocity v from an initial velocity zero
• s can be obtained from the third equation of motion: v2 - u2 = 2as
Which gives: s = (v2-u2)⁄2a.
9. So Ek = F×s = m × a × (v2-u2)⁄2a = m(v2-u2)⁄2
But initial velocity u = 0
So we get: Ek = m(v2)⁄2.
We can write in the form of an equation for Kinetic energy:
In the next section, we will see Potential energy.
We will see energy from a scientific point of view.
1. Consider a fast moving cricket ball.
• If it hits a wicket, the stumps will be thrown away. The stumps got displaced.
• That is., the ball did some work on the stumps. For doing that work, the ball must possess some energy.
• After hitting the stumps, the energy of the ball is lost. It can no longer displace another wicket.
2. Another example: A heavy iron piece is raised to a height of about 80 cm above the ground
and let go.
• It falls upon a nail driven one fourth into a wooden piece.
• When the iron falls on the nail, the nail is driven upto three fourth into the wood
• Here the iron piece did some work on the nail. Because it displaced the nail.
• But once the nail is driven to three fourth, the energy is lost. That is., energy is all used up.
• If the iron was raised to a height more that 80 cm, it may have had sufficient energy to drive the nail completely into the wooden piece.
3. One more example: A toy car is wound and then placed on the floor.
• It starts to move. So the spring inside the car is doing some work.
• That means, the spring has some energy.
• When all that energy is used up, the toy car stops.
■ In the first example, if the cricket ball was stationary, it would not be able to displace the wicket.
• That is., it would not be able to do work
• That is., it would not have possessed any energy■ In the second example, if the iron piece was on the ground, it would not be able to drive the nail
• That is., it would not be able to do work
• That is., it would not have possessed any energy
■ In the third example, if the spring was not wound up, the toy would not have moved.
• That is., it would not be able to do work
• That is., it would not have possessed any energy
■ Just as we specified a method to measure work, we need a method to measure energy also.
Let us see how energy can be measured:
• Consider an object like cricket ball, stone etc.,
♦ If it is stationary, it cannot do any work.
♦ If it is moving with some velocity, it can do some work
♦ If it is moving with a greater velocity, it can do greater work
♦ If it is moving with a lesser velocity, it can do only a lesser work
• That means, motion of a body gives it some energy.
■ The energy possessed by a body due to it's motion is called kinetic energy.
Let us summarise:
1. A moving body can do work
2. The ability to do that work comes from the energy that the body possess
3. The energy is due to the motion of the body
4. This energy possessed by a moving body is called kinetic energy
Now we want to know 'how much' kinetic energy that the body possess. Let us try to find it mathematically:
1. If the body is at rest, it does not have kinetic energy.2. If it is moving with a velocity v, it has a certain amount of kinetic energy. Let us denote it as Ek
• It acquired this Ek because it acquired a velocity v
3. But the body cannot acquire a velocity v all by itself. Some one or some thing has to accelerate it from velocity zero to velocity v
4. If the body is accelerated, it is clear that some one or some thing applied a force F on the body.
• Because, Force = mass × acceleration. (Details here)
5. Now, if the body is moving with a velocity v, it is clear that, the body is being displaced
So at this stage, we know the following 4 facts:
(i) The body was at rest
(ii) It was acted upon by a force F
(iii) As a result, it was subjected to acceleration and acquired a velocity v
(iv) While increasing the velocity from zero to v, the body would have moved through a certain distance.
6. We call this distance as: 'displacement'.
• Let s be the displacement in which the velocity v was acquired
7. Then we can say this: The force F caused a displacement of s
• That means, the force F did some work (F×s) on the body. (Details in the previous section)
• This work will remain in the body moving with velocity v. And so, this work is equal to the kinetic energy.
• Thus, F×s is the kinetic energy possessed by the moving body.
♦ We can write: Ek = F×s
♦ We have to calculate this F×s. Let us try:
8. We have two terms: F and s
• F can be written as (m×a).
♦ Where m is the mass of the body and
♦ a is the acceleration which enabled the body to acquire a velocity v from an initial velocity zero
• s can be obtained from the third equation of motion: v2 - u2 = 2as
Which gives: s = (v2-u2)⁄2a.
9. So Ek = F×s = m × a × (v2-u2)⁄2a = m(v2-u2)⁄2
But initial velocity u = 0
So we get: Ek = m(v2)⁄2.
We can write in the form of an equation for Kinetic energy:
We see that, the 'energy possessed' is equal to the 'work done'. So work and energy has the same unit N m or joule
Now we will see some solved examples
Solved example 4.3
An object of mass 15 kg is moving with a uniform velocity of 4 m s-1 . What is the kinetic energy possessed by the object?
Solution:
• Mass of the object, m = 15 kg
• Velocity, v = 4
• We have: Ek = 1⁄2 mv2 = 1⁄2 × 15 × 42 = 120 J
• So kinetic energy of the object is 120 J
Solved example 4.4
What is the work to be done to increase the velocity of a car from 30 kmph to 60 kmph if the mass of the car is 1500 kg?
Solution:
1. First we need to convert the velocities from kmph to m s-1.
• u = 30 kmph = 30 × 1000⁄3600 = 8.33 m s-1.
• v = 60 kmph = 60 × 1000⁄3600 = 16.67 m s-1.
2. The car would be initially at rest. Some work has to be done to make the car move with a velocity of 30 kmph. We know how to calculate this work. It is given by:
Ek = 1⁄2 mv2 = 1⁄2 × 1500 × 8.332 = 52041.68 J
3. Similarly, starting from rest, the energy required to make the car move with a velocity of 16.67 kmph is given by:
Ek = 1⁄2 mv2 = 1⁄2 × 1500 × 16.672 = 208416.68 J
4. But an energy of 52041.68 J was already present in the car because, it was moving with a velocity of 8.33 m s-1.
5. So the extra energy required to change the velocity from 8.33 m s-1 to 16.67 m s-1 is equal to:
208416.68 - 52041.68 = 156375 J
Solved example 4.5
The kinetic energy of an object of mass, m moving with a velocity of 5 m s-1 is 25 J. What will be its kinetic energy when its velocity is doubled? What will be its kinetic energy when its velocity is increased three times?
Solution:
• We have: Ek = 1⁄2 mv2 . So we can write: 25 = 1⁄2 × m × 52 ⇒ 25 = 1⁄2 × m × 25 ⇒ m = 2 kg
Case (i)
1. When the velocity is doubled, we have: v = 10 m s-1
2. So new Ek = 1⁄2 mv2 = 1⁄2 × 2 × 102 = 100 J
3. 100 is 4 times 25. So, when the velocity in doubled, Ek becomes 4 times. This is because 22 = 4
Case (ii)
1. When the velocity is increased 3 times, we have: v = 15 m s-1
2. So new Ek = 1⁄2 mv2 = 1⁄2 × 2 × 152 = 225 J
3. 225 is 9 times 25. So, when the velocity in increased 3 times, Ek becomes 9 times. This is because 32 = 9
• Mass of the object, m = 15 kg
• Velocity, v = 4
• We have: Ek = 1⁄2 mv2 = 1⁄2 × 15 × 42 = 120 J
• So kinetic energy of the object is 120 J
Solved example 4.4
What is the work to be done to increase the velocity of a car from 30 kmph to 60 kmph if the mass of the car is 1500 kg?
Solution:
1. First we need to convert the velocities from kmph to m s-1.
• u = 30 kmph = 30 × 1000⁄3600 = 8.33 m s-1.
• v = 60 kmph = 60 × 1000⁄3600 = 16.67 m s-1.
2. The car would be initially at rest. Some work has to be done to make the car move with a velocity of 30 kmph. We know how to calculate this work. It is given by:
Ek = 1⁄2 mv2 = 1⁄2 × 1500 × 8.332 = 52041.68 J
3. Similarly, starting from rest, the energy required to make the car move with a velocity of 16.67 kmph is given by:
Ek = 1⁄2 mv2 = 1⁄2 × 1500 × 16.672 = 208416.68 J
4. But an energy of 52041.68 J was already present in the car because, it was moving with a velocity of 8.33 m s-1.
5. So the extra energy required to change the velocity from 8.33 m s-1 to 16.67 m s-1 is equal to:
208416.68 - 52041.68 = 156375 J
Solved example 4.5
The kinetic energy of an object of mass, m moving with a velocity of 5 m s-1 is 25 J. What will be its kinetic energy when its velocity is doubled? What will be its kinetic energy when its velocity is increased three times?
Solution:
• We have: Ek = 1⁄2 mv2 . So we can write: 25 = 1⁄2 × m × 52 ⇒ 25 = 1⁄2 × m × 25 ⇒ m = 2 kg
Case (i)
1. When the velocity is doubled, we have: v = 10 m s-1
2. So new Ek = 1⁄2 mv2 = 1⁄2 × 2 × 102 = 100 J
3. 100 is 4 times 25. So, when the velocity in doubled, Ek becomes 4 times. This is because 22 = 4
Case (ii)
1. When the velocity is increased 3 times, we have: v = 15 m s-1
2. So new Ek = 1⁄2 mv2 = 1⁄2 × 2 × 152 = 225 J
3. 225 is 9 times 25. So, when the velocity in increased 3 times, Ek becomes 9 times. This is because 32 = 9
In the next section, we will see Potential energy.
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