Question

A wheel rotates with an angular acceleration \(\alpha_{z}\) given by $$ \alpha_{z}=4 a t^{3}-3 b t^{2} $$ where \(t\) is the time and \(a\) and \(b\) are constants. If the wheel has an initial angular velocity \(\omega_{0}\), write the equations for \((a)\) the angular velocity and \((b)\) the angle turned through as functions of time.

Short answer

The angular velocity as a function of time is given by \(\omega = at^{4} - bt^{3} + \omega_0\) and the angle turned as a function of time is given by \(\theta = \frac{a}{5}t^{5} - \frac{b}{4}t^{4} + \omega_0t\).

Step-by-step solution

Identifying the quantities and interpreting the problem

We are given that the wheel has an initial angular velocity denoted by \(\omega_{0}\), with an angular acceleration \(\alpha_{z}\) denoted by the equation \(\alpha_{z}=4 a t^{3}-3 b t^{2}\). The task is to calculate the angular velocity and the angle turned.

Calculate Angular Velocity

To find the angular velocity (\(\omega\)), we need to integrate the angular acceleration (\(\alpha_z\)) with time. This is because velocity is the time integral of acceleration. Integrating \(\alpha_{z}=4 a t^{3}-3 b t^{2}\) with respect to time (t) gives \(\int (4 a t^{3}-3 b t^{2})dt = at^{4} - bt^{3} + C\), where C is the constant of integration. Since at t = 0, \(\omega = \omega_0\), the integration constant C will be equal to \(\omega_0\). Hence, the equation of angular velocity as a function of time is given by \(\omega = at^{4} - bt^{3} + \omega_0\)

Calculate Angle Turned

To find the angle turned (\(\theta\)), we integrate the angular velocity (\(\omega\)) with time. This is because displacement (or in this case angle turned) is the time integral of velocity. Integrating \(\omega = at^{4} - bt^{3} + \omega_0\) with respect to time (t) gives \(\int (at^{4} - bt^{3} + \omega_0)dt = \frac{a}{5}t^{5} - \frac{b}{4}t^{4} + \omega_0t + D\), where D is the constant of integration. Since at t = 0, \(\theta = 0\), the integration constant D will be equal to zero. Hence, the equation for the angle turned as a function of time is given by \(\theta = \frac{a}{5}t^{5} - \frac{b}{4}t^{4} + \omega_0t\)

Key concepts

Angular Acceleration

Angular acceleration is a measure of how quickly the angular velocity of a rotating object changes with time. It is the rotational equivalent of linear acceleration. In the given problem, the angular acceleration \( \alpha_{z} \) is expressed as a function of time \( t \), specifically \( \alpha_{z} = 4 a t^{3} - 3 b t^{2} \), where \( a \) and \( b \) are constants.
The higher the angular acceleration, the quicker the object speeds up or slows down its rotation.

• An object with positive angular acceleration speeds up its rotation.
• If the angular acceleration is negative, the object slows down.

This concept is crucial in understanding the dynamics of rotational motion. It helps in predicting how a rotating object will move over time. Knowing this, we can calculate angular velocity and other rotational characteristics.

Angular Velocity

Angular velocity refers to how fast an object rotates or revolves around a fixed point or axis. The angular velocity in this exercise is influenced by the given angular acceleration.
To find the angular velocity equation, we integrate the angular acceleration \( \alpha_{z} \) over time. The integration involves collecting the time-dependent terms and solving:\[\omega = \int (4 a t^{3} - 3 b t^{2}) \, dt = a t^{4} - b t^{3} + C \]where \( C \) is the constant of integration. Given the initial angular velocity \( \omega_0 \) at time \( t = 0 \), we find that \( C = \omega_0 \). Therefore, the angular velocity as a function of time is:\[\omega = a t^{4} - b t^{3} + \omega_0\]This equation helps us understand how the speed of rotation changes with time, starting from the initial state defined by \( \omega_0 \).

Integral Calculus

Integral calculus is a fundamental part of understanding motion, both linear and rotational. It allows us to determine accumulated quantities like velocity and displacement from their rates of change, acceleration.
In our exercise, integral calculus helps us transition from angular acceleration to angular velocity, and then from angular velocity to the angle turned.

• Finding Angular Velocity: We integrate the given angular acceleration \( \alpha_{z} \) with respect to time.
• Calculating Angle Turned: We integrate the angular velocity \( \omega \) to find the angle \( \theta \).

The integration process adds the dimensions of change over time, providing equations that describe how these quantities evolve, reflecting the physical progression of motion.

Kinematics

Kinematics is the study of motion without considering the forces that cause it. This branch of mechanics deals with the description of motion in terms of capacities like position, velocity, and acceleration.
Rotational kinematics, as in this context, deals with rotating bodies. It utilizes concepts such as angular velocity and angular acceleration.

• It emphasizes how these quantities describe the rotation of objects over time.
• Kinematics mostly concerns itself with the mathematical models that describe motion.

In this problem, we're looking at the kinematics of a rotating wheel by determining how the angular velocities and positions change over time due to a known angular acceleration. This illustration helps depict how objects behave in various rotational movements.