Question

A simple pendulum is at rest. After sometime v velocity is given to the bob of the pendulum. So the potential energy at the highest point will be? (m is mass of bob, h is maximum height)
1. mgh
2. \( {1 \over 2}mv^2\)
3. \( {1 \over 2}mv^2\) + mgh
4. ​\( {1 \over 2}mv^2\)​ - mgh

Answer 1

Correct Answer - Option 2 : \( {1 \over 2}mv^2\)

CONCEPT:​

• Mechanical energy: An energy due to its position and motion i.e. Sum of Potential energy and kinetic energy.

• The Kinetic Energy of an object due to its linear speed is given by:

​​E = 1/2 (m × v2) 

where m is the mass of a body and v is the speed.

• Potential Energy or Gravitation Potential Energy of an object is given by:

PE = m g h

where m is the mass of a body, g is gravitational acceleration, and h is the height from the earth's surface.

• Conservation of mechanical energy: The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative

Total initial mechanical energy = Total final mechanical energy

EXPLANATION:

• Given that v velocity is given to the pendulum when it is at rest.
• At this moment it will have kinetic energy

K​​E = 1/2 (m × v2) 

Let this is the base point i.e. at this point of pendulum h = 0

So potential energy = mgh = 0

• At the highest point, there will be no speed. So there will be kinetic energy. Hence there will be only potential energy.
  • ​Since there is the only gravitational force acting on the given system which is a conservative force. So total mechanical energy will be conserved.
  • So initial mechanical energy will be equal to final mechanical energy.
• So the kinetic energy at the lowest point will be converted into potential energy at the highest point.

The potential energy at highest point = kinetic energy at the lowest point

The potential energy at highest point = \( {1 \over 2}mv^2\)

• So the correct answer is option 2.