Question

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

Angular velocity is 0 and acceleration is \(-\Theta \omega^2\) at \(\Theta\). At \(\Theta/2\), velocity is \(-\Theta \omega \frac{\sqrt{3}}{2}\) and acceleration is \(-\Theta \omega^2 \frac{1}{2}\).

Step-by-step solution

Understand the Given Parameters

The balance wheel of a watch oscillates with an angular amplitude \( \Theta \), angular frequency \( \omega \), and phase angle \( \phi = 0 \). We need to find expressions for angular velocity \( \frac{d\theta}{dt} \) and angular acceleration \( \frac{d^2\theta}{dt^2} \) as functions of time. These quantities depend on the oscillatory motion described by \( \theta(t) = \Theta \cos(\omega t + \phi) \). Since \( \phi = 0 \), \( \theta(t) = \Theta \cos(\omega t) \).

Calculate Angular Velocity

To find angular velocity \( \frac{d\theta}{dt} \), differentiate \( \theta(t) = \Theta \cos(\omega t) \) with respect to time \( t \). The derivative of \( \cos(\omega t) \) is \( -\omega \sin(\omega t) \). Thus,\[\frac{d\theta}{dt} = -\Theta \omega \sin(\omega t).\]

Calculate Angular Acceleration

Find angular acceleration \( \frac{d^2\theta}{dt^2} \) by differentiating the expression for angular velocity, \( \frac{d\theta}{dt} = -\Theta \omega \sin(\omega t) \). The derivative of \( \sin(\omega t) \) is \( \omega \cos(\omega t) \). Thus,\[\frac{d^2\theta}{dt^2} = -\Theta \omega^2 \cos(\omega t).\]

Evaluate Angular Velocity at Maximum Displacement \( \Theta \)

At angular displacement \( \theta = \Theta \), the term inside the cosine is zero, \( \omega t = 0 \). From \( \frac{d\theta}{dt} = -\Theta \omega \sin(\omega t) \), since \( \sin(0) = 0 \), the angular velocity is 0. So, the angular velocity is zero when the displacement is at a maximum.

Evaluate Angular Acceleration at Maximum Displacement \( \Theta \)

At angular displacement \( \theta = \Theta \), substitute in \( \omega t = 0 \) for \( \frac{d^2\theta}{dt^2} = -\Theta \omega^2 \cos(\omega t) \). Since \( \cos(0) = 1 \), the angular acceleration is:\[\frac{d^2\theta}{dt^2} = -\Theta \omega^2.\]

Evaluate Kinematic Parameters at Half Displacement \( \Theta/2 \)

For angular displacement \( \theta = \Theta/2 \), we solve \( \Theta \cos(\omega t) = \Theta/2 \), which gives \( \cos(\omega t) = 1/2 \). This occurs at \( \omega t = \pi/3 \). Evaluating angular velocity \( \frac{d\theta}{dt} = -\Theta \omega \sin(\omega t) \), with \( \sin(\pi/3) = \sqrt{3}/2 \), gives\[\frac{d\theta}{dt} = -\Theta \omega \frac{\sqrt{3}}{2}.\]For angular acceleration \( \frac{d^2\theta}{dt^2} = -\Theta \omega^2 \cos(\omega t) \), plugging \( \cos(\pi/3) = 1/2 \) gives\[\frac{d^2\theta}{dt^2} = -\Theta \omega^2 \frac{1}{2}.\]

Key concepts

Oscillatory Motion

Oscillatory motion refers to a repetitive back-and-forth movement over the same path, much like a swing on a playground or a pendulum in a clock. It's characterized by its regularity and periodic nature, meaning the motion occurs at regular intervals. In physics, this can be seen in systems like pendulums, springs, or any system that returns to an equilibrium state after being disturbed.

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

Angular velocity is a measure of how quickly an object rotates or revolves relative to another point, often the center of a circle. It tells us the rate of change of angular displacement (\(\theta\)). In mathematical terms, it's expressed as the derivative of angular displacement with respect to time.

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

Angular acceleration is the rate of change of angular velocity over time. It provides insights into how quickly an object's rotational speed is changing at any given moment. Critical in oscillatory systems, angular acceleration helps us understand the dynamics of motion.

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

The phase angle, often denoted as \(\phi\), indicates the initial angle of the system at time \(t = 0\), influencing the starting point of the oscillation. The phase angle can shift the entire wave left or right on a graph.

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

Trigonometric functions, such as sine and cosine, are pivotal in describing oscillatory motion. They capture the periodic nature of motion through their wave patterns.

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.

These functions showcase the inherent symmetry and predictability in oscillatory systems. They help visualize how objects move back and forth as part of a broader cycle, crucial for accurately predicting future motion dynamics.