Question

A simple pendulum is suspended from the ceiling of a stationary elevator, and the period is determined. (i) When the elevator accelerates upward, is the period (a) greater, (b) smaller, or (c) unchanged? (ii) When the elevator has a downward acceleration, is the period (a) greater, (b) smaller, or (c) unchanged? (iii) When the elevator moves with constant upward velocity, is the period of the pendulum (a) greater, (b) smaller, or (c) unchanged?

Short answer

(i) The elevator accelerates upward the period is small; Hence option (b) is the correct answer for this question.

(ii)The elevator has a downward acceleration the period is greater; Hence option (a) is the correct answer for this question.

(iii) The elevator moves with constant upward velocity the period of the pendulum is unchanged; hence option (c) is the correct answer for this question.

Step-by-step solution

Relationship between the period and acceleration

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 when the elevator accelerates upward

Fromstep(1),we have

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

The upward acceleration has the same effect as an increased gravitational acceleration.

The elevator accelerates upward the period is small; Hence option (b) is the correct answer for this question.

Step 3(ii): Find the period when the elevator has a downward acceleration

The downward acceleration has the same effect as a decreased gravitational acceleration.

The elevator has a downward acceleration the period is greater; Hence option (a) is the correct answer for this question.

Step 4(iii): Find the period when the elevator moves with constant upward velocity

The absence of acceleration means that the effective gravitational field is the same as that for a stationary elevator.

The elevator moves with constant upward velocity the period of the pendulum is unchanged; Hence option (c) is the correct answer for this question.