## Key Concepts:

- Gravitational potential energy
- The law of conservation energy

### Introduction:

The objects in the images possess potential energy because of their special shape and position.

When the special shape and position of the objects are disturbed, the objects come into motion. The energy possessed by the objects in motion is kinetic energy. In all these examples transfer of energy is taking place the potential energy is getting converted into the kinetic energy of the object.

The law of conservation of energy states that energy is neither created nor destroyed, but changes from one form to another.

**Explanation:**

Let’s take an example of an apple on a tree at a height of (H) from the ground.

- The gravitational force is pulling the apple down.
- The apple in the image possesses only potential energy. It has no kinetic energy as it is at rest.
- The energy of this apple is due to its position above the ground, so this energy is known as gravitational potential energy.

**Expression for total mechanical energy**

**Case 1: An apple hanging from a tree **

Mass of the apple = m

Height of the apple above the ground = H

Potential energy = gravitational force x height above the ground

P.E. = mg x H

Potential Energy = mgH

Kinetic energy = ( the apple at rest speed= v = 0)

Total energy = Potential energy + Kinetic energy

Total energy = mgH + 0

T.E. = P.E. = mgH

**Case 2: An apple falling from a tree **

Mass of the apple = m

Height of the apple above the ground = h

Potential energy = gravitational force x height above the ground

P.E. = mgh

Initial velocity of the apple = u = 0

velocity at height h = v

Acceleration = a = g

Displacement = (H-h)

Kinetic energy = 1/2mv^{2} – 1/2mu^{2} = 1/2mv^{2} ——(1)

From 3rd equation of motion v^{2} – u^{2} = 2g(H-h)

v^{2} = 2g(H-h) ——(2)

From eq. (1) and (2) Kinetic energy = mg(H-h)

Total mechanical energy = P.E. + K.E. =mgh + mg(H-h) = mgH

**Case 3: An apple about to reach the ground **

Mass of the apple = m

Height of the apple above the ground = 0

Potential energy = gravitational force x height above the ground

P.E. = mgH = 0

Initial velocity of the apple = u = 0

velocity when it is about to touch the ground = v

Acceleration = a = g

Displacement = H

Kinetic energy = 1/2mv^{2} – 1/2mu^{2} = 1/2mv^{2} ——(1)

From 3rd equation of motion v^{2} – u^{2} = 2gH

v^{2} = 2gH ——(2)

From eq. (1) and (2) Kinetic energy = mgH

Total mechanical energy = P.E. + K.E. = 0 + mgH = mgH

**Conclusion:**

When an apple falls freely from a tree during its entire path the total mechanical energy of the apple remains conserved only the transformation of energy is taking place.

Mechanical energy is the sum of kinetic energy and potential energy, i.e.,

Mechanical energy = potential energy + kinetic energy

Mechanical energy = mgH = Constant

Mechanical energy is the sum of kinetic energy and potential energy of the body.

M.E = K.E + P. E

I.e., Mechanical energy = potential energy + kinetic energy

M.E = mgh + ½ mv^{2}

**Example of the law of conservation of energy**

**:** A roller coaster ride.

At the top of the roller coaster, the rider has lots of potential energy because the cart is at a larger height above the ground. When the cart starts to fall it starts gaining kinetic energy. During the ride, the rider keeps losing and gaining height. Gaining height will create potential energy while losing height will create kinetic energy.

**Conclusion:**

- Potential energy is the stored energy due to height and kinetic energy is the energy due to motion.
- Potential energy is highest at the top of the ride and kinetic energy is highest at the bottom of the ride.
- In absence of friction, the rider hit the same height on the opposite side of the track.
- The bar graph and pie chart show the relation between kinetic energy and potential energy. As potential energy increases the kinetic energy decreases and vice versa.
- Potential energy + Kinetic energy = Total mechanical energy
- When we change the mass, friction, and gravity they affect the rider’s energy.
- When mass increases the rider’s energy also increases.
- In presence of frictional force, the rider slowly loses his energy and finally stops.
- When gravity increases the rider’s energy also increases and he repeats his periodic motion more quickly.

**Question 1:**

On a roller coaster, a rider has a 1250 J of kinetic energy at an instant, and his mechanical energy is 3000 J. (Take g as 10 m/sec^{2})

- Find the potential energy of the rider.
- If the rider has a mass of 25kg, what is his height above the ground at that instant?
- What is the speed of the rider at that instant?

**Solution:**

The mass of the rider = m = 25 kg

The kinetic energy of the rider at an instant = K.E. = 1250 J

The total mechanical energy of the rider = M.E. = 3000 J

We know from the law of conservation of energy

Total mechanical energy = Potential energy + Kinetic energy

M.E. = P.E. + K.E.

5000 J = P.E. + 1250 J

**Potential energy = 1750 J**

P.E. = mgh

1750 = 25 x 10 x h, Height = h = 7 m

**The height of the rider above the ground = 7m**

K.E. = 1/2 mv^{2 } = 1250 = 1/2 x 25 x v^{2}

v = 10 m/sec

**The speed of the rider at that instant = 10 m/sec**

**Question 2:**

A ball of mass 200gm thrown vertically up from the ground reaches a maximum height of 20m in 10s. Find the potential energy of the ball. (Take g as 10 m/sec^{2})

- 40000 J
- 20 J
- 20000 J
- 40 J

**Solution:**

Potential energy = P.E. = mgh

Mass = 200 gm

Height = h = 20m

P.E. = (200/1000)kg x 10 m/sec^{2} x 20m

P.E. = 40 J

Thus, the potential energy of the ball = 40 J

**Question 3:**

An apple of mass 25gm is falling from a tree, find the kinetic energy of the apple when it is about to reach the ground from a height of 50m. (Take g as 10 m/sec^{2})

- 12.5 J
- 1.25 J
- 1250 J
- 250 J

**Solution:**

Mass of the apple= m = 25gm

Height = H = 50m

In this problem total mechanical energy = Constant

P.E. of the apple on the tree = K.E. of the apple when it is about to reach the ground

K.E. = P.E. = mgH

K.E. = (25/1000)x (10 m/sec^{2}) x 50m

K.E. = 12.5 J

Thus, the kinetic energy of the apple when it is just about to reach the ground is 12.5 J.

## Summary:

- Law of conservation of energy states that energy is neither created nor destroyed but changes from one form to another.
- In absence of any resistive forces like air drag, friction, etc. The total mechanical energy remains constant.
- Mechanical energy is the sum of kinetic energy and potential energy of the body.
- Le, Mechanical energy = potential energy + kinetic energy

ME = KE +P. E = Constant

ME. = mgh + 1/2 mv^{2} = Constant

- We have seen examples when an apple falls freely from a tree and a person on a roller coaster ride in both situations during the entire path the total mechanical energy of the apple and the person remain conserved only the transformation of energy is taking place.

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