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 49a

Chapter 14, Problem 49a

A clock pendulum oscillates at a frequency of 2.5 Hz. At t = 0, it is released from rest starting at an angle of 12° to the vertical. Ignoring friction, what will be the position (angle in radians) of the pendulum at t = 0.25 s?

Verified step by step guidance

Step 1: Recognize that the motion of the pendulum can be modeled as simple harmonic motion (SHM) for small angles. The angular position θ(t) as a function of time is given by the equation:

θ (t)= θ

₀cos(2πft), where

θ

₀ is the initial angular displacement,

f

is the frequency, and

t

is the time.

Step 2: Convert the initial angle from degrees to radians, as the equation requires the angle in radians. Use the conversion formula:

θ

₀=θ₀(deg)×(π/180). Here,

θ

₀(deg)=12°.

Step 3: Substitute the given values into the SHM equation. The frequency

f

is 2.5 Hz, and the time

t

is 0.25 s. The equation becomes:

θ (t)= θ

₀cos(2π(2.5)(0.25)).

Step 4: Simplify the argument of the cosine function. Calculate the product inside the cosine:

2 π (2.5)(0.25)

. This will give the phase angle in radians.

Step 5: Evaluate the cosine function using the simplified phase angle and multiply it by the initial angular displacement

θ

₀ (in radians). This will yield the angular position

θ (t)

t =0.25

seconds.

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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 describes the oscillatory motion of systems like pendulums, where the restoring force is proportional to the displacement from the equilibrium position. In this case, the pendulum's motion can be modeled as SHM, characterized by a sinusoidal function that describes its position over time.

Angular Displacement

Angular displacement refers to the angle through which an object has rotated about a specific axis, measured in radians. For the pendulum, the initial angle of 12° must be converted to radians (approximately 0.209 radians) to analyze its position at any given time during its oscillation.

Frequency and Period

Frequency is the number of oscillations per second, while the period is the time taken for one complete cycle of motion. The frequency of 2.5 Hz indicates that the pendulum completes 2.5 cycles every second, allowing us to calculate the period (T = 1/f) and determine the pendulum's position at specific time intervals.

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