Ch 14: Periodic Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 14: Periodic Motion

Problem 16a

Chapter 14, Problem 16a

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is 0.090 m, it takes the block 2.70 s to travel from x = 0.090 m to x = -0.090 m. If the amplitude is doubled, to 0.180 m, how long does it take the block to travel from x = 0.180 m to x = -0.180 m?

Verified step by step guidance

Understand that the block is undergoing Simple Harmonic Motion (SHM), which is characterized by oscillations around an equilibrium position. The motion is described by the equation:

x = A cos (ω t)

, where

A

is the amplitude,

ω

is the angular frequency, and

t

is the time.

Determine the period of the motion. The time given for the block to travel from

x = 0.090 m

x = - 0.090 m

is half the period of the motion, because it represents a full oscillation from one extreme to the other. Therefore, the period

T

is twice the given time:

T = 2.70 \times 2

Recognize that the period of SHM is independent of the amplitude. This means that even if the amplitude is doubled, the period remains the same. Therefore, the time taken for the block to travel from

x = 0.180 m

x = - 0.180 m

is still half the period.

Calculate the time for the block to travel from

x = 0.180 m

x = - 0.180 m

using the period calculated in step 2. Since this is half the period, the time is

2.70

seconds.

Conclude that the time taken for the block to travel from

x = 0.180 m

x = - 0.180 m

is the same as the time taken for the smaller amplitude, due to the properties of SHM.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion refers to the oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. In SHM, the motion is periodic, and the system oscillates around an equilibrium position. The time period and frequency are constant, regardless of amplitude changes.

Amplitude in SHM

Amplitude in SHM is the maximum displacement from the equilibrium position. It determines the energy of the system but does not affect the period or frequency of the motion. Doubling the amplitude increases the total energy but the time taken for a complete cycle remains unchanged, as the period is independent of amplitude.

Period of Oscillation

The period of oscillation is the time taken for one complete cycle of motion in SHM. It is determined by the mass and the spring constant in a spring-mass system and remains constant regardless of changes in amplitude. Thus, the time to travel from one extreme to the other is half the period, unaffected by amplitude changes.

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Related Practice

Textbook Question

A 0.400-kg object undergoing SHM has a_x = -1.80 m/s² when x = 0.300 m. What is the time for one oscillation?

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Textbook Question

A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 N/m. At t = 0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 m/s. Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.

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Textbook Question

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is 0.090 m, it takes the block 2.70 s to travel from x = 0.090 m to x = -0.090 m. If the amplitude is doubled, to 0.180 m, how long does it take the block to travel from x = 0.090 m to x = -0.090 m?

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Textbook Question

Weighing Astronauts. This procedure has been used to 'weigh' astronauts in space: A 42.5-kg chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes 1.30 s to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair takes 2.54 s for one cycle. What is the mass of the astronaut?

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Textbook Question

The point of the needle of a sewing machine moves in SHM along the x-axis with a frequency of 2.5 Hz. At t = 0 its position and velocity components are +1.1 cm and -15 cm/s, respectively. Find the acceleration component of the needle at t = 0.

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Textbook Question

An object is undergoing SHM with period 0.900 s and amplitude 0.320 m. At t = 0 the object is at x = 0.320 m and is instantaneously at rest. Calculate the time it takes the object to go (a) from x = 0.320 m to x = 0.160 m. (b) from x = 0.160 m to x = 0.

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