Problem 7 - University Physics with Modern Physics

Question

The angle \(\theta\) through which a disk drive turns is given by \(\theta(t)=a+b t-c t^{3},\) where \(a, b,\) and \(c\) are constants \(t\) is in seconds, and \(\theta\) is in radians. When \(t=0, \theta=\pi / 4\) rad and the angular velocity is \(2.00 \mathrm{rad} / \mathrm{s},\) and when \(t=1.50 \mathrm{s},\) the angular acceleration is 1.25 \(\mathrm{rad} / \mathrm{s}^{2}\) , (a) Find \(a, b,\) and \(c,\) including their units. b) What is the angular acceleration when \(\theta=\pi / 4\) rad? (c) What are \(\theta\) and the angular velocity when the angular acceleration is 3.50 \(\mathrm{rad} / \mathrm{s}^{2} ?\)

Step-by-Step Solution

Answer

(a) \( a = \frac{\pi}{4} \) rad, \( b = 2.00 \) rad/s, \( c \approx 0.1389 \) rad/s³. (b) \( \alpha = 0 \) rad/s². (c) \( \theta \approx 6.36 \) rad, \( \omega \approx -4.67 \) rad/s.

Step 1: Understanding Given Equation

The angle through which the disk drive turns is represented by the equation \( \theta(t) = a + bt - ct^3 \). Here, \( \theta \) is the angle (in radians), \( t \) is the time (in seconds), and \( a, b, c \) are constants to be determined.

Step 2: Use Initial Condition for \(\theta\)

At \( t = 0 \), the angle \( \theta = \frac{\pi}{4} \) radians. Substitute \( t = 0 \) into the equation: \[ a + b(0) - c(0)^3 = \frac{\pi}{4} \] This simplifies to \( a = \frac{\pi}{4} \).

Step 3: Determine Angular Velocity Equation

The angular velocity \( \omega(t) \) is the first derivative of \( \theta(t) \) with respect to \( t \). Thus,\[ \omega(t) = \frac{d}{dt}(a + bt - ct^3) = b - 3ct^2 \].

Step 4: Use Initial Condition for Angular Velocity

Given that \( \omega(0) = 2.00 \) rad/s at \( t = 0 \), substitute into the angular velocity equation: \[ \omega(0) = b - 3c(0)^2 = 2.00 \] This simplifies to \( b = 2.00 \) rad/s.

Step 5: Formulate Angular Acceleration Equation

The angular acceleration \( \alpha(t) \) is the derivative of the angular velocity \( \omega(t) \). \[ \alpha(t) = \frac{d}{dt}(b - 3ct^2) = -6ct \].

Step 6: Use Condition for Angular Acceleration

At \( t = 1.50 \) s, \( \alpha = 1.25 \) rad/s². Substitute into the angular acceleration equation: \[ -6c(1.50) = 1.25 \] Solve for \( c \): \[ c = -\frac{1.25}{-6 \times 1.50} = -\frac{1.25}{-9} \] \( c \approx 0.1389 \) rad/s³.

Step 7: Find Angular Acceleration for Part (b)

When \( \theta = \frac{\pi}{4} \) rad, \( t = 0 \) based on our initial conditions. Using \( \alpha(t) = -6ct \), substitute \( t = 0 \): \[ \alpha(0) = -6c(0) = 0 \] The angular acceleration is \( 0 \) rad/s².

Step 8: Determine \( \theta \) and \( \omega \) for Part (c)

When \( \alpha = 3.50 \) rad/s², \[ -6ct = 3.50 \] Substitute \( c = 0.1389 \): \[ -6(0.1389)t = 3.50 \] Find \( t \): \[ t = -\frac{3.50}{-6 \times 0.1389} \approx 4.20 \] seconds.Substitute \( t \approx 4.20 \) into \( \theta(t) \): \[ \theta(4.20) = \frac{\pi}{4} + 2.00(4.20) - (0.1389)(4.20)^3 \approx 6.36 \] radians.Substitute into \( \omega(t) \): \[ \omega(4.20) = 2.00 - 3(0.1389)(4.20)^2 \approx -4.67 \] rad/s.

Key Concepts

Angular Velocity

Angular velocity is a fundamental concept in rotational motion. It describes how quickly an object rotates or revolves around a central point. In mathematical terms, angular velocity is defined as the rate of change of the angular displacement with respect to time. It tells us how much angle an object covers per unit of time and is usually denoted by the symbol \( \omega \).

Angular velocity can be calculated by differentiating the angle position function \( \theta(t) \). So, if the angle through which something rotates is given by a function, say \( \theta(t) = a + bt - ct^3 \), then its angular velocity would be the first derivative of this function with respect to time:

• \( \omega(t) = \frac{d\theta(t)}{dt} = b - 3ct^2 \)

This equation tells us how the angular velocity changes as time progresses. Here, the terms \( b \) and \(-3ct^2 \) determine the initial angular velocity and how it changes over time, respectively. It's important to understand that even though angular velocity can change over time, it encapsulates the idea of rotational speed at any specific moment.

In practical applications, knowing the angular velocity helps in analyzing how fast a disk or any other rotational system is spinning. It also allows us to calculate other parameters like angular displacement and the effects of forces acting on rotating systems.

Angular Acceleration

Angular acceleration describes the rate at which angular velocity changes with time. It is a critical concept in understanding the dynamics of rotating objects. When an object spins faster or slower, its angular velocity changes, which introduces a new measure – angular acceleration, typically represented by the symbol \( \alpha \).

The angular acceleration is found by differentiating the angular velocity equation. For example, if you have an equation for angular velocity, such as \( \omega(t) = b - 3ct^2 \), then the angular acceleration is:

• \( \alpha(t) = \frac{d\omega(t)}{dt} = -6ct \)

This formula reveals that angular acceleration is dependent on the parameter \( c \) and time \( t \). If \( c \) is positive, the acceleration indicates the increasing speed of rotation, while a negative \( c \) points to rotation slowing down.

Understanding angular acceleration is crucial when assessing how forces affect rotational motion. For example, applying a constant force would result in uniform angular acceleration, directly influencing the object's rotational speed. In various applications, especially in mechanical and electronic devices, controlling the angular acceleration is key for operational efficiency.

Kinematics of Rotational Motion

Kinematics is the branch of physics that deals with the motion of objects without considering the forces causing the motion. In rotational motion, kinematics is particularly concerned with how rotational position, velocity, and acceleration relate to each other.

The key quantities are:

• Angular displacement \( \theta \)
• Angular velocity \( \omega \)
• Angular acceleration \( \alpha \)

These elements work together to describe rotational motion fully. For a rotating disk with angular displacement \( \theta(t) = a + bt - ct^3 \), understanding how each parameter affects the object's motion involves calculating \( \omega \) from \( \theta \), and \( \alpha \) from \( \omega \).

The relationships among these quantities can be summarized by the kinematic equations:

• Angular velocity as the first derivative of angular displacement: \( \omega(t) = \frac{d\theta(t)}{dt} \)
• Angular acceleration as the derivative of angular velocity: \( \alpha(t) = \frac{d\omega(t)}{dt} \)

Kinematic equations help predict future motion characteristics based on initial conditions. With initial angular velocity and known angular acceleration, one can determine the motion at any point in time, vital for planning in various engineering contexts.

Understanding rotational kinematics allows scientists and engineers to effectively design and control systems involving wheels, gears, and rotating machinery, thus playing a significant role in the development and improvement of technology.