Question

A simple pendulum has a period of 2.5 s. (i) what is its period if its length is made four times larger? (a) 1.25 s (b) 1.77 s (c) 2.5 s (d) 3.54 s (e) 5 s (ii) What is its period if the length is held constant at its initial value and the mass of the suspended bob is made four times larger? Choose from the same possibilities.

Short answer

(a)The period if its length is made four times larger is$T_f=2T_i=2\times 2.5s=5s$, Hence option (e) is the correct answer for this question.

(b) The period if the length is held constant at its initial value and the mass of the suspended bob is made four times larger is$T_f=T_i=2.5s$, Hence option (c) is the correct answer for this question.

Step-by-step solution

Relationship between the length and time 

A simple pendulum of length L can be modeled to move in simple harmonic motion for small angular displacements from the vertical. Its period is

$T=2\pi \sqrt{\frac{L}{g}}$

$T=$Period of oscillation

$L=$Length of pendulum

$g=$Gravitational acceleration

Step 2(i): Find the period if its length is made four times larger

Fromstep(1),we have

$T_i=2\pi \sqrt{\frac{L_i}{g}}$and $T_f=2\pi \sqrt{\frac{L_f}{g}}$, where i and f stands for initial and final respectively.

$$\begin{gathered} T_f=2\pi \sqrt{\frac{4L_i}{g}} \\ {T}_f=2.2\pi \sqrt{\frac{L_i}{g}} \\ {T}_f=2T_i \end{gathered}$$

The period becomes larger by a factor of 2, to become 5 s.

Hence option (e) is the correct answer for this question.

Step 3(ii): Find the period if the length is held constant at its initial value and the mass of the suspended bob is made four times larger

Changing the mass has no effect on the period of a simple pendulum.

Hence option (c) is the correct answer for this question.