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Chapter 12: Problem 43

A simple pendulum of length \(l\) is suspended through the ceiling of an elevator. Find the time period of small oscillations if the elevator (a) is going up with an acceleration \(a_{0}\) (b) is going down with an acceleration \(a_{0}\) and (c) is moving with a uniform velocity.

Short Answer

Expert verified
(a) \( T = 2\pi \sqrt{\frac{l}{g + a_0}} \), (b) \( T = 2\pi \sqrt{\frac{l}{g - a_0}} \), (c) \( T = 2\pi \sqrt{\frac{l}{g}} \).

Step by step solution

01

Understanding the Basic Pendulum Formula

The time period of a simple pendulum in a stationary elevator is given by the formula \( T = 2\pi \sqrt{\frac{l}{g}} \), where \( g \) is the acceleration due to gravity, and \( l \) is the length of the pendulum.
02

Pendulum in an Upward Accelerating Elevator

When the elevator moves upward with an acceleration \( a_0 \), the effective acceleration acting on the pendulum becomes \( g + a_0 \). The new time period is \( T = 2\pi \sqrt{\frac{l}{g + a_0}} \).
03

Pendulum in a Downward Accelerating Elevator

When the elevator moves downward with acceleration \( a_0 \), the effective acceleration becomes \( g - a_0 \). This yields a new time period of \( T = 2\pi \sqrt{\frac{l}{g - a_0}} \).
04

Pendulum in an Elevator with Uniform Velocity

If the elevator moves with uniform velocity, the acceleration is zero. Thus, the time period remains \( T = 2\pi \sqrt{\frac{l}{g}} \) as there are no additional forces acting on the pendulum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Pendulum
A simple pendulum is a classic physics demonstration that consists of a weight, known as a bob, attached to the end of a string or rod of a certain length. The other end of the string is fixed at a point, allowing the bob to swing freely back and forth. The beauty of a simple pendulum is its straightforward mechanisms:
  • The motion is primarily governed by the forces of tension in the string and gravity acting on the bob.
  • As it swings, it traces out an arc of a circle, oscillating around its equilibrium position.
  • It is an idealized version, assuming no air resistance and a non-stretchable, inextensible string.
This simplicity makes it perfect for studying fundamental concepts such as harmonic motion and periodicity in physics.
Time Period of Oscillation
The time period of oscillation refers to the time it takes for the pendulum to complete one full swing, returning to its original position after moving to the extreme in one direction and back. For a simple pendulum, the time period is calculated using the formula \[ T = 2\pi \sqrt{\frac{l}{g}} \]where:
  • \( T \) is the time period, measured in seconds.
  • \( l \) is the length of the pendulum's string, measured in meters.
  • \( g \) is the acceleration due to gravity, approximately \(9.81 \text{ m/s}^2\) on Earth's surface.
The formula shows that:
  • Only the length of the pendulum and the gravitational pull affect the period.
  • The mass of the bob and the amplitude of the swing do not affect the period for small oscillations, making pendulums predictable and reliable for measuring time.
Accelerating Reference Frame
When we consider pendulum motion within a non-static environment, such as an accelerating elevator, we're dealing with an accelerated reference frame. This concept modifies the forces affecting the pendulum's motion:
  • In an upward accelerating frame, the effective gravity increases, leading to a faster oscillation.
  • Conversely, in a downward accelerating frame, the effective gravity decreases, which slows down the oscillation.
This is because the acceleration of the frame adds to or subtracts from the real gravitational acceleration, altering how forces act on the pendulum. Thinking of these situations as happening in an 'altered gravity' helps conceptualize how the pendulum behaves in different scenarios.
Effective Gravity
Effective gravity refers to the apparent gravitational force experienced in a non-inertial frame, like that of an accelerating elevator. It is the combination of the actual gravitational force and any additional forces due to the acceleration of the reference frame.For a simple pendulum:
  • When an elevator moves upwards with an acceleration \(a_0\), the effective acceleration is \( g + a_0 \), increasing the pull experienced by the pendulum.
  • Conversely, when the elevator descends with acceleration \(a_0\), the effective acceleration is \( g - a_0 \), reducing the pendulum's perceived gravitational force.
  • If the elevator moves at a constant velocity, there are no additional forces, and the effective gravity remains \( g \).
This mechanistic adjustment in perceived forces emphasizes why the pendulum's period changes depending on its movement, explaining the broader physics of pendulum dynamics in practical applications.

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Most popular questions from this chapter

The angle made by the string of a simple pendulum with the vertical depends on time as \(\theta=\frac{\pi}{90} \sin \left[\left(\pi \mathrm{s}^{-1}\right) t\right]\). Find the length of the pendulum if \(g=\pi^{2} \mathrm{~m} \mathrm{~s}^{-2}\).

A block suspended from a vertical spring is in equilibrium. Show that the extension of the spring equals the length of an equivalent simple pendulum, i.e., a pendulum having frequency same as that of the block.

A hollow sphere of radius \(2 \mathrm{~cm}\) is attached to an \(18 \mathrm{~cm}\) long thread to make a pendulum. Find the time period of oscillation of this pendulum. How does it differ from the time period calculated using the formula for a simple pendulum?

A small block of mass \(m\) is kept on a bigger block of mass \(M\) which is attached to a vertical spring of spring constant \(k\) as shown in the figure. The system oscillates vertically. (a) Find the resultant force on the smaller block when it is displaced through a distance \(x\) above its equilibrium position. (b) Find the normal force on the smaller block at this position. When is this force smallest in magnitude ? (c) What can be the maximum amplitude with which the two blocks may oscillate together?

A simple pendulum is constructed by hanging a heavy ball by a \(5-0 \mathrm{~m}\) long string. It undergoes small oscillations. (a) How many oscillations does it make per second? (b) What will be the frequency if the system is taken on the moon where acceleration due to gravitation of the moon is \(1.67 \mathrm{~m} \mathrm{~s}^{-2}\) ?
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