Question

Assume that a 4.00 kg pendulum bob is hanging from a 7.90 m long cable. The cable is attached to a hook in the ceiling 10 m up from the floor. (a) If a person releases the bob, from the small angle approximation, how long will it take the bob to swing back to him? s (b) What is meant by the small angle approximation? any angle less than 3° any angle less than 5° any angle less than 10° any angle less than 15°

Answer 1

Answer:

Explanation:

Given

mass of Pendulum m=4 kg

Length of cable= 7.90 m

Distance between roof and Floor=10 m

$mg\sin\theta$will Provide Torque

thus $mg\sin \theta \times L=I\cdot \alpha$

where I=moment of inertia

$\alpha$=angular acceleration

here $\theta$is small therefore \sin \theta \approx \theta

$\frac{g}{L}\theta =\alpha$

thus $\omega ^2=\frac{g}{L}$

$\omega =\sqrt{\frac{g}{L}}$

And $T\cdot \omega =2\pi$

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

$T=2\pi \sqrt{\frac{7.90}{9.8}}=5.64 s$

Small angle refers to any angle where

$\sin \theta \approx \theta$i.e. any angle less than $15^{\circ}$

Answer 2

The time taken for bob to swing back to him is mathematically given as

T=5.64 s

Small angle approximation refers to any angle less than 15°, Option D

What is the time taken for bob to swing back to him and what is small angle approximation?

Question Parameter(s):

A 4.00 kg pendulum bob is hanging from a 7.90 m long cable.

The cable is attached to a hook in the ceiling 10 m up

Generally, the equation for the angular velocity  is mathematically given as

$\omega =\sqrt{\frac{g}{L}}$

Thereofre

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

$T=2\pi \sqrt{\frac{7.90}{9.8}}$

T=5.64 s

In conclusion,  small angle approximation speaks to any angle that satisfies

$Sin\theta =\theta$

Therefore, any angle less than 15°, Option D

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