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Chapter 10: Problem 57

A \(0.50-\mathrm{kg}\) mass is suspended from a string, forming a pendulum. The period of this pendulum is 1.5 s when the amplitude is \(1.0 \mathrm{cm} .\) The mass of the pendulum is now reduced to \(0.25 \mathrm{kg} .\) What is the period of oscillation now, when the amplitude is \(2.0 \mathrm{cm} ?\) (W tutorial: change in period)

Short Answer

Expert verified
The pendulum has an initial mass of 0.50 kg and an initial amplitude of 1.0 cm, with a period of 1.5 s. Answer: The period of oscillation for the new mass and amplitude will remain the same as the initial period, which is 1.5 s.

Step by step solution

01

Recall the period formula for a pendulum

The period of oscillation of a pendulum is given by the formula: $$ T = 2π\sqrt{\frac{l}{g}} $$ Where \(T\) is the period, \(l\) is the length of the pendulum's string, and \(g\) is the acceleration due to gravity (approximately \(9.81 m/s^2\)).
02

Analyze the effect of changing mass and amplitude on the period

It's important to notice that the period formula of a pendulum doesn't involve mass and amplitude. Thus, changes in mass or amplitude don't affect the period as long as the pendulum remains in the regime of simple harmonic motion (where the amplitude is small, typically less than 20° from the vertical).
03

Calculate period with given values

Given that the period of the pendulum with a \(0.50-kg\) mass and \(1.0 cm\) amplitude is \(1.5 s\), we already have the period of oscillation under these conditions: $$ T_1 = 1.5~s $$
04

Determine the period of oscillation with the new mass and amplitude

Since we've established that mass and amplitude changes don't affect the period, we can directly say that the period of oscillation when the mass is reduced to \(0.25-kg\) and the amplitude is doubled to \(2.0 cm\) will still be the same as before: $$ T_2 = T_1 = 1.5 s $$ So, the period of oscillation remains the same as 1.5 s for the new mass and amplitude of the pendulum.

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