## The concept of Conical Pendulum:

• A conical pendulum consists of a bob of mass ‘m’ revolving in a horizontal circle with constant speed ‘v’ at the end of a string of length ‘l’. In this case, the string makes a constant angle with the vertical.
• In this case, the bob of pendulum describes a horizontal circle and the string describes a cone.

### Expression for Period of Conical Pendulum:

• A conical pendulum consists of a bob of mass ‘m’ revolving in a horizontal circle with constant speed ‘v’ at the end of a string of length ‘l’. The string makes a constant angle ‘θ’ with the vertical. let ‘h’ be the depth of the bob below the support.
• The tension ‘F’ in the string can be resolved into two components. Horizontal ‘Fsin θ’ and vertical ‘Fcos θ’. The vertical component (F cos θ) balances the weight mg of the vehicle.

• The horizontal component ‘F sin’ provides the necessary centripetal force.

Dividing equation (1) by (2) we get,

But v = rω
Where ω is angular speed and T is the period of the pendulum.

This is an expression for the time period of a conical pendulum.

### Expression for Tension in the String of Conical Pendulum:

Squaring equations (1) and (2) and adding

This is an expression for the tension in the string of a conical pendulum.

### Note:

• A simple pendulum is a special case of a conical pendulum in which angle made by the string with vertical is zero i.e. θ = 0°.
• The period of simple pendulum is given by

Note: Above explanation is based on assumption that reference frame is inertial.

#### Period of a simple pendulum in Non-Inertial Reference Frame:

• Reference frames which are at rest or moving with constant velocity with respect to the earth are called inertial reference frames.
• Reference frames which are moving with acceleration with respect to the earth are called non-inertial reference frames. In case of the non-inertial reference frame, we have to consider the pseudo force acting on the bob of the pendulum and corresponding changes should be done in the formula.
• For  a simple pendulum oscillating in a vehicle moving horizontally with acceleration (a) the time period  is

The pendulum will make an angle θ = tan-1(g/a)  with the vertical

• For  a simple pendulum oscillating in a lift moving upward with acceleration (a) the time period of oscillation is

• For  a simple pendulum oscillating in a lift moving downward with acceleration (a) the time period of oscillation is

• For  a conical pendulum oscillating in a vehicle moving horizontally with acceleration (a) the time period  is

The pendulum will make an angle θ + tan-1(g/a)  with the vertical

• For  a conical pendulum oscillating in a lift moving upward with acceleration (a) the time period of oscillation is

• For  a conical pendulum oscillating in a lift moving downward with acceleration (a) the time period of oscillation is