If the mass of the bob of a simple pendulum is increased, then which of the following statement is correct:

CONCEPT:

Simple pendulum:

• When a point mass is attached to an inextensible string and suspended from fixed support then it is called a simple pendulum.
• The time period of a simple pendulum is defined as the time taken by the pendulum to finish one complete oscillation.

\(⇒ T = 2\pi\sqrt{\frac{L}{g}}\)

• The above formula is only valid for small angular displacements.
• Since the motion of the bob is along a circle of length L and centre at the support point, the radial acceleration is given as,

⇒ aR = ω2L

• The radial acceleration is provided by the net radial force T' - mg.cos θ.
• The tangential acceleration is provided by mg.sinθ.
• Torque τ about the support is entirely provided by the tangential component of a force and it is given as,

⇒ τ = -L.(mg.sin θ)

• The negative sign shows that this is the restoring torque that tends to reduce angular displacement.

Where, T = Time period of oscillation, L = length of the pendulum, m = mass of the bob, θ = angle made by the string with the vertical axis and g = gravitational acceleration

EXPLANATION:

• We know that the time period of a simple pendulum is given as,

\(⇒ T = 2\pi\sqrt{\frac{L}{g}}\)     -----(1)

• By equation 1 it is clear that the time period of a simple pendulum is independent of the mass of the bob.
• So when the mass of the bob of a simple pendulum is increased, its time period will not change.
• We know that the restoring torque τ for a simple pendulum about the support is given as,

⇒ τ = -L.(mg.sin θ)

⇒ τ ∝ m     -----(2)

Where, T = Time period of oscillation, L = length of the pendulum, m = mass of the bob, θ = angle made by the string with the vertical axis and g = gravitational acceleration

• By equation 2, it is clear that the restoring torque of a simple pendulum is directly proportional to the mass of the bob.
• Therefore if the mass of the bob of a simple pendulum is increased, then its restoring torque will also increase. Hence, option 3 is correct.

Additional Information

• The amplitude of a simple pendulum is defined as the maximum distance traveled by the pendulum from the equilibrium position to one side.
• The length of a simple pendulum is defined as the distance between the point of suspension to the center of the bob.
• The tangential acceleration is given as,

\(\Rightarrow \alpha=-\frac{mgL}{I}\sinθ=-\frac{g}{L}\sinθ\)

Where I = moment of inertia, and θ = angle made by the string with the vertical

• Now if θ is small, sinθ can be approximated by θ, and the tangential acceleration is given as,

\(\Rightarrow \alpha=-\frac{mgL}{I}θ=-\frac{g}{L}θ\)