Problem 5 - University Physics with Modern Physics

Question

A child is pushing a merry-go-round. The angle through which the merry-go- round has turned varies with time according to \(\theta{(t) = \gamma t + \beta t^3}\), where \(\gamma =\) 0.400 rad/s and \(\beta =\) 0.0120 rad/s\(^3\). (a) Calculate the angular velocity of the merry-go-round as a function of time. (b) What is the initial value of the angular velocity? (c) Calculate the instantaneous value of the angular velocity \(\omega$$_z\) at \(t =\) 5.00 s and the average angular velocity \(\omega_{av-z}\) for the time interval \(t =\) 0 to \(t =\) 5.00 s. Show that \(\omega_{av-z}\) is not equal to the average of the instantaneous angular velocities at \(t =\) 0 and \(t =\) 5.00 s, and explain.

Step-by-Step Solution

Answer

(a) \( \omega(t) = 0.400 + 3 \times 0.0120 \times t^2 \). (b) 0.400 rad/s. (c) Instantaneous: 1.300 rad/s, Average: 0.7 rad/s. These differ due to non-linear acceleration.

Step 1: Expression for Angular Velocity

The angular velocity \( \omega(t) \) is the derivative of the angle \( \theta(t) \) with respect to time. Thus, we have: \[ \omega(t) = \frac{d\theta}{dt} = \frac{d}{dt} (\gamma t + \beta t^3) \]Differentiating gives:\[ \omega(t) = \gamma + 3 \beta t^2 \]

Step 2: Initial Angular Velocity

To find the initial angular velocity, substitute \( t = 0 \) into the expression for \( \omega(t) \):\[ \omega(0) = \gamma + 3 \beta (0)^2 = \gamma = 0.400 \, \text{rad/s} \]

Step 3: Instantaneous Angular Velocity at t = 5 s

Substitute \( t = 5.00 \,s \) into the expression for \( \omega(t) \) to find the instantaneous angular velocity:\[ \omega(5) = 0.400 + 3(0.0120)(5)^2 \]Calculate:\[ \omega(5) = 0.400 + 3 \times 0.0120 \times 25 = 0.400 + 0.900 = 1.300 \, \text{rad/s} \]

Step 4: Average Angular Velocity Over Interval

The average angular velocity \( \omega_{av-z} \) over the interval from \( t = 0 \) to \( t = 5 \) is given by:\[ \omega_{av-z} = \frac{\theta(5) - \theta(0)}{5 - 0} \]Calculate \( \theta(5) \):\[ \theta(5) = 0.400 \times 5 + 0.0120 \times (5)^3 = 2.0 + 1.5 = 3.5 \, \text{radians} \] Since \( \theta(0) = 0 \),\[ \omega_{av-z} = \frac{3.5 - 0}{5} = 0.7 \, \text{rad/s} \]

Step 5: Compare Averages

The average of the instantaneous angular velocities at \( t = 0 \) and \( t = 5 \) is:\[ \frac{\omega(0) + \omega(5)}{2} = \frac{0.400 + 1.300}{2} = 0.850 \, \text{rad/s} \]This differs from \( \omega_{av-z} = 0.7 \, \text{rad/s} \) because the average angular velocity is affected by the acceleration term \( \beta t^3 \), which changes the rate of rotation non-linearly over time.

Key Concepts

Understanding Rotational Kinematics

Rotational kinematics is the branch of physics that deals with the motion of objects that rotate or spin. Objects like merry-go-rounds rotate around an axis, and rotational kinematics helps describe this motion.

• Key terms include angular position, angular velocity, and angular acceleration.
• The equations and concepts are similar to those used in linear kinematics but are adapted for rotation.

By knowing the angular position of an object over time, we can calculate its angular velocity and acceleration, giving us a complete picture of its rotational motion.

Exploring Angular Displacement

Angular displacement refers to the change in the angle through which an object has rotated, measured in radians. It helps us understand how much rotation has occurred.For a merry-go-round, if the initial angle is zero, the angular displacement can be calculated using the function \[ \theta(t) = \gamma t + \beta t^3 \]where \( \gamma \) and \( \beta \) are coefficients that represent the constant and timedependent components of the motion, respectively.

• This equation integrates time to determine how the angle changes as the object turns.
• Understanding angular displacement helps in predicting future positions and in calculating other dynamic factors like velocity and acceleration.

Instantaneous Velocity Clarified

Instantaneous velocity is the speed of an object at a specific moment in time. For rotational motion, this is called angular velocity and is denoted as \( \omega(t) \).The angular velocity indicates how quickly the object is rotating at any given point, and can be derived by differentiating the expression for angular displacement:\[ \omega(t) = \frac{d\theta}{dt} = \gamma + 3\beta t^2 \]

• This formula reveals that angular velocity changes with time due to the term \( 3\beta t^2 \).
• At \( t = 5\) seconds, for instance, this function gives us the precise rotational speed at that moment, helping gauge performance or synchronization in mechanical systems.

Calculating Average Velocity

Average angular velocity over a period gives a broad picture of how an object has been rotating, smoothing out its speed variations into a single value.This is calculated by the change in angular displacement over time:\[ \omega_{av-z} = \frac{\theta(5) - \theta(0)}{5} \]

• This formula computes the average rate of rotation from start to finish of a given time span.
• Crucially, this value can differ from the mean of instantaneous velocities at the start and end times due to the nature of non-uniform motion.

Understanding both instantaneous and average angular velocities allows us to analyze rotational motion in different contexts, from simple machinery to complex engineering systems.