Chapter 14: Problem 44
The balance wheel of a watch vibrates with an angular amplitude \(\Theta\), angular frequency \(\omega\), and phase angle \(\phi =\) 0. (a) Find expressions for the angular velocity \(d\theta/dt\) and angular acceleration \(d^2\theta/dt^2\) as functions of time. (b) Find the balance wheel's angular velocity and angular acceleration when its angular displacement is \(\Theta\), and when its angular displacement is \(\Theta\)/2 and \(\theta\) is decreasing. (\(Hint\): Sketch a graph of \(\theta\) versus \(t\).)
Short Answer
Step by step solution
Understand the Given Parameters
Calculate Angular Velocity
Calculate Angular Acceleration
Evaluate Angular Velocity at Maximum Displacement \( \Theta \)
Evaluate Angular Acceleration at Maximum Displacement \( \Theta \)
Evaluate Kinematic Parameters at Half Displacement \( \Theta/2 \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Oscillatory Motion
For example, in the case of a balance wheel in a watch, the wheel experiences oscillatory motion as it vibrates back and forth. This motion can be described mathematically using a sine or cosine function, which accounts for its periodic behavior. Here, the angular displacement function is given as \[\theta(t) = \Theta \cos(\omega t)\] where \(\Theta\) is the angular amplitude, reflecting the maximum displacement from the equilibrium position. This function gives us insight into how the wheel moves over time and is key to understanding how oscillatory systems function in general.
Angular Velocity
Specifically, for oscillatory motion described by \(\theta(t) = \Theta \cos(\omega t)\), its angular velocity is calculated by differentiating this function with respect to time:\[\frac{d\theta}{dt} = -\Theta \omega \sin(\omega t)\]
The result shows that angular velocity is dependent on the sine of the angular frequency multiplied by time. This expression indicates how the speed of rotation varies over time, reaching its maximum magnitude when the displacement is zero and slowing down to zero when displacement reaches a maximum.
Angular Acceleration
From the expression of angular velocity, \(\frac{d\theta}{dt} = -\Theta \omega \sin(\omega t)\), angular acceleration can be derived through further differentiation:\[\frac{d^2\theta}{dt^2} = -\Theta \omega^2 \cos(\omega t)\]
This result indicates that angular acceleration depends on a cosine function of time and is proportional to the square of the angular frequency. It shows that when angular displacement reaches its maximum or minimum value, the acceleration is at peak magnitude, exerting a restoring force that pulls the system back toward equilibrium.
Phase Angle
In the case of simple harmonic motion like that of a balance wheel with \(\phi = 0\), this simplifies the angular displacement function:\[\theta(t) = \Theta \cos(\omega t)\]
A phase angle of zero means that the balance wheel starts its motion at its maximum displacement. The absence of a phase shift makes it straightforward to analyze the system's motion since its sinusoidal behavior begins at the peak of its cycle, directly related to the amplitude.
Trigonometric Functions
For the balance wheel example, the functions \(\cos(\omega t)\) and \(\sin(\omega t)\) help us describe the changing displacement, velocity, and acceleration over time:
- \(\cos(\omega t)\) describes how displacement changes, varying from maximum to minimum in a smooth, predictable manner.
- \(\sin(\omega t)\) is used to represent angular velocity, highlighting how velocities change throughout the cycle.