Which of the following combinations of variables results in the greatest period for a pendulum? length = L, mass = M, and maximu

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Which of the following combinations of variables results in the greatest period for a pendulum? length = L, mass = M, and maximu

m angular displacement = degrees length = L, mass = M, and maximum angular displacement = 3 degrees length = 2L, mass = M/2, and maximum angular displacement = 1 degree length = 1.5L, mass = 2M, and maximum angular displacement = 2 degrees length = L, mass = 4M, and maximum angular displacement = 4 degrees

Physics

1 answer:

Yuliya22 [3.3K] · Answer 1 · 2026-04-01T13:34:21+03:00

Response:

length = 2L, mass = M/2, and maximum angular displacement = 1 degree

Clarification:

We examine only small amplitude oscillations (as in this scenario), which keeps the angle θ sufficiently small. In such situations, it's important to note that the pendulum's motion can be described by the equation:

$\ddot{\theta}=\frac{g}{l}\theta$

The resulting solution is:

$\theta=Asin(\omega t + \phi)$

Here, $\omega=\sqrt\frac{g}{l}$ represents the angular frequency of the oscillations, enabling us to find the period:

$T=\frac{2\pi}{\omega}\\T=2\pi\sqrt\frac{l}{g}$

As a result, the period of a pendulum is determined solely by its length and is independent of both its mass and angle, provided the angle remains small. Therefore, the choice with the longest length gives the longest period.