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Chapter 13: Problem 50

Derive the period of a simple pendulum by considering the horizontal displacement \(x\) and the force acting on the bob, rather than the angular displacement and torque.

Short Answer

Expert verified
The period of the pendulum is given by \(T = 2\pi \sqrt{L / g}\).

Step by step solution

01

Identify Variables

Let's denote the length of the pendulum as \(L\), the mass of the bob as \(m\), the horizontal displacement as \(x\), the gravitational force as \(Fg = mg\), where \(g\) is the gravitational acceleration, and the restoring force as \(F\). The angle made by the pendulum with the vertical is \(\theta\).
02

Apply Newton's Second Law

According to Newton's second law, the restoring force \(F\) is equal to the mass of the object times its acceleration \(a\), i.e., \(F = ma\). For a pendulum, this force is provided by the weight component perpendicular to the tension or \(F = mg \sin(\theta)\). If we consider small angle approximation, we can replace \(\sin(\theta)\) with \(\theta\). As the displacement \(x = L \theta\), we can express \(\theta\) as \(x/L\). So, \(F = mg (x/L)\), or \(a = g (x/L)\).
03

Use Simple Harmonic Motion Equation

In simple harmonic motion, acceleration is proportional to the displacement but is in the opposite direction, which can be written as \(a = - \omega^2 x\), with negative sign indicating the force is restoring. Here, \(\omega\) is the angular frequency. By comparing with the expression obtained in Step 2, we find \(\omega^2 = g/L\). Angular frequency and period \(T\) are related as \(T = 2\pi / \omega\).
04

Calculate the Pendulum Period

Substitute the value of \(\omega\) using the relation \(\omega^2 = g/L\) into the period equation. Then the period \(T\) becomes \(T = 2\pi \sqrt{L / g}\).

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Most popular questions from this chapter

an ornithologist who knows you've studied physics. She asks you for a noninvasive way to measure birds" masses. You propose using a bird feeder in the shape of a 50 -cm-diameter disk of mass \(340 \mathrm{~g}\), suspended by a wire with torsional constant \(5.00 \mathrm{~N} \cdot \mathrm{m} / \mathrm{rad}(\mathrm{Fig} .13 .35)\). Two birds land on opposite sides and the feeder goes into torsional oscillation at \(2.6 \mathrm{~Hz}\). Assuming the birds have the same mass, what is it?

A \(600-\mathrm{g}\) block on a frictionless, horizontal surface is attached to a rather limp spring with \(k=8.7 \mathrm{~N} / \mathrm{m}\). A second block rests on the first, and the whole system executes simple harmonic motion with period \(2.1 \mathrm{~s}\). When the amplitude of the motion is increased to \(35 \mathrm{~cm}\), the upper block just begins to slip. What's the coefficient of static friction between the blocks?

Explain why the frequency of a damped system is lower than that of the equivalent undamped system.

A hummingbird's wings vibrate at about \(45 \mathrm{~Hz}\). What's the corresponding period?

Why is critical damping desirable in a car's suspension?
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