Ch. 14 - Oscillations

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 14 - Oscillations

Problem 16

Chapter 14, Problem 16

Determine the phase constant ϕ in Eq. 14–4 if, at t = 0, the oscillating mass is at 𝓍 = ― 1/2 A.

Verified step by step guidance

Start by recalling the equation for simple harmonic motion:

x(t) = A cos(ωt + ϕ)

, where

x(t)

is the displacement,

A

is the amplitude,

ω

is the angular frequency,

t

is time, and

ϕ

is the phase constant.

t = 0

, the equation simplifies to

x(0) = A cos(ϕ)

. This is because

ωt

becomes zero, leaving only the phase constant

ϕ

in the cosine function.

The problem states that at

t = 0

, the displacement

x(0) = -\(\frac{1}{2}\)A

. Substitute this value into the simplified equation:

-\(\frac{1}{2}\)A = A cos(ϕ)

Divide through by

A

(assuming

A \(\neq\) 0

) to isolate the cosine term:

cos(ϕ) = -\(\frac{1}{2}\)

To find

ϕ

, take the inverse cosine (arccos) of

-\(\frac{1}{2}\)

. Remember that cosine is negative in the second and third quadrants, so

ϕ

will have two possible solutions:

ϕ = \(\pi\) - \(\frac{\pi}{3}\)

ϕ = \(\pi\) + \(\frac{\pi}{3}\)

. Simplify these expressions to determine the phase constant.

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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 is a type of periodic motion where an object oscillates around an equilibrium position. The motion can be described by a sine or cosine function, characterized by parameters such as amplitude, angular frequency, and phase constant. Understanding SHM is crucial for analyzing oscillatory systems, as it provides the foundational framework for predicting the position and velocity of the mass at any given time.

Amplitude (A)

Amplitude is the maximum displacement of an oscillating object from its equilibrium position. In the context of SHM, it represents the peak value of the oscillation, indicating how far the mass moves from the center point. The amplitude is a key factor in determining the energy of the oscillation, as greater amplitudes correspond to higher potential and kinetic energy during the motion.

Phase Constant (ϕ)

The phase constant, denoted as ϕ, is a parameter in the equations of SHM that determines the initial position of the oscillating object at time t = 0. It shifts the sine or cosine function along the time axis, allowing for the description of various starting conditions. In this problem, finding the phase constant is essential to accurately describe the motion of the mass when it is at a specific position, such as x = -1/2 A.