Question

A simple pendulum with mass m = 1.8 kg and length L = 2.69 m hangs from the ceiling. It is pulled back to an small angle of θ = 8.7° from the vertical and released at t = 0.

1) What is the period of oscillation?

2) What is the magnitude of the force on the pendulum bob perpendicular to the string at t=0?

3) What is the maximum speed of the pendulum?

4) What is the angular displacement at t = 3.65 s? (give the answer as a negative angle if the angle is to the left of the vertical)

5) What is the magnitude of the tangential acceleration as the pendulum passes through the equilibrium position?

6) What is the magnitude of the radial acceleration as the pendulum passes through the equilibrium position?

7) Which of the following would change the frequency of oscillation of this simple pendulum?

increasing the mass

decreasing the initial angular displacement

increasing the length

hanging the pendulum in an elevator accelerating downward

Answer 1

1)

Formula Used :

Time period of a Simple Pendulum,

----------------eqn (1)

Solution :

Here, L = 2.69 m , g = 9.8 ms^-2

Putting these values in eqn (1), we get

2)

Diagram :

Concept : Here tension, T is provided by Centipetal force,

On putting values of m, t and L we get,

T = 17.6 N

Force on the pendulum bob perpendicular to the string at t=0 will be,

On putting these values from above, we get

3)

Concept : maximum speed of the pendulum will be at mean position i.e at A

Formula :

On putting these values from above, we get

4)

Here, , ,

On putting these values from above, we get,

angular displacement with vertical,

7) Answer :

increasing the length,

hanging the pendulum in an elevator accelerating downward

Reason:

Formula Used:

Clearly, Time Period depends on

i. effective length of pendulum (L), i.e by increasing the length, and

ii. effective acceleration due to gravity (g) i.e by hanging the pendulum in an elevator accelerating downward (which decreases g)