Question

A simple pendulum, while oscillating, rises to a maximum vertical height of 5 cm from its rest position when it reaches to its extreme position on one side. If mass of the bob of the simple pendulum is 500 g and g = 10 ms?, find => the velocity of bob at its mean position.

Answer 1

Answer:

This question can be solved by using Energy Conservation Principle.

Given that the pendulum rises to a height of 5 cm from it's rest or mean position. Let's take the distance of rest position to ground is 0 cm. Also it is given that,

• Mass = 500 g
• Acceleration due to gravity = 10 m/s²

According to the Energy Conservation Principle,

⇒ Total Energy at point A = Total Energy at point B.

⇒ Total Energy = Potential Energy + Kinetic Energy

Let the mean position be Point A, and the maximum vertical position be Point B. Calculating Total Energy at Point A and B, we get:

⇒ Potential Energy at Point A = mgh

⇒ Potential Energy = 0.5 kg × 10 m/s² × 0 m = 0 J

⇒ Kinetic Energy at Point A = 0.5 × mv²

⇒ Kinetic Energy = 0.5 × 0.5 kg × v²

⇒ Kinetic Energy = 0.25 v²   ...(i)

Hence Total Energy at Point A = P.E + K.E

⇒ Total Energy = 0 + 0.25 v² = 0.25 v²

Now considering Point B we get:

⇒ Potential Energy = mgh

⇒ Potential Energy = 0.5 kg × 10 m/s² × 0.05 m

⇒ Potential Energy = 0.25 J

Since the velocity is zero at the extreme position, the value of Kinetic Energy would be 0 J. Hence the total energy at Point B is:

⇒ Total Energy at Point B = 0.25 J   ...(ii)

Equating (i) and (ii) we get:

⇒ 0.25 v² = 0.25 J

⇒ v² = 0.25/0.25

⇒ v² = 1

⇒ v = ± 1

⇒ v = 1 m/s (Ignoring negative value)

Hence the velocity of bob at it's mean position is 1 m/s.

Answer 2

Given :-

A simple pendulum, while oscillating, rises to a maximum vertical height of 5 cm from its rest position when it reaches its extreme position on one side. If the mass of the bob of the simple pendulum is 500 g and g = 10 m/s²

To Find :-

Find the velocity of the bob at its mean position.

Solution :-

1 kg = 1000 kg

500 g = 500/1000

500 g = 0.5 kg

In Case 1

• Mass = 0.5 kg
• Height = 0 m
• Acceleration due to gravity = 10 m/s²

Kinetic energy = 1/2 × m × v²

Kinetic energy = 1/2 × 0.5 × v²

Kinetic energy = 0.5/2 × v²

Kinetic energy = 5/20 × v²

Kinetic energy = v²/4

Potential energy = mgh

Potential energy = 0.5 × 10 × 0

Potential energy = 5/10 × 10 × 0

Potential energy = 5 × 0

Potential energy = 0 J

Total energy = Kinetic energy + Potential energy

Total energy = 1v²/2 + 0

Total energy = v²/2

In Case 2

Height = 5 cm = 5/100 = 0.05 m

• Mass = 0.5 kg
• Height = 0.05 m

• Acceleration due to gravity = 10 m/s²

Kinetic energy = 1/2 × m × v²

Kinetic energy = 1/2 × 0.5 × (0)²

Kinetic energy = 0.5/2 × 0

Kinetic energy = 0.25 × 0

Kinetic energy = 0 J

Potential energy = mgh

Potential energy = 0.5 × 10 × 0.05

Potential energy = 5/10 × 10 × 0.05

Potential energy = 0.25 J

So,

0.25v² = 0.25

v² = 0.25/0.25

v² = 25/25

v² = 1

v = √1

v = 1

Hence

velocity of bob at its mean position is 1 m/s