Problem 19 - College Physics

Question

Emilie's potter's wheel rotates with a constant 2.25 \(\mathrm{rad} / \mathrm{s}^{2}\) angular acceleration. After 4.00 \(\mathrm{s}\) , the wheel has rotated through an angle of 60.0 rad. What was the angular velocity of the wheel at the beginning of the 4.00 s interval?

Step-by-Step Solution

Answer

The initial angular velocity was 10.5 rad/s.

Step 1: Identify Known Variables

We need to solve for Emilie's initial angular velocity \( \omega_0 \). The known values are angular acceleration \( \alpha = 2.25 \, \text{rad/s}^2 \), time \( t = 4.00 \, \text{s} \), and final angular displacement \( \theta = 60.0 \, \text{rad} \).

Step 2: Use the Angular Motion Formula

The equation for angular displacement when starting with an initial angular velocity and constant angular acceleration is: \[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \] Substitute the known values: \[ 60.0 = \omega_0 \times 4.00 + \frac{1}{2} \times 2.25 \times (4.00)^2 \]

Step 3: Simplify the Equation

Calculate the term that involves the angular acceleration: \[ \frac{1}{2} \times 2.25 \times (4.00)^2 = 18 \] So, substitute back: \[ 60.0 = \omega_0 \times 4.00 + 18 \]

Step 4: Solve for Initial Angular Velocity

Rearrange the equation to solve for \( \omega_0 \): \[ \omega_0 \times 4 = 60.0 - 18 \] \[ \omega_0 = \frac{42}{4} = 10.5 \, \text{rad/s} \]

Key Concepts

Angular Velocity

Angular velocity refers to how fast an object rotates around a central point or axis. Unlike linear velocity, which measures how fast something moves in a straight direction, angular velocity measures rotation. It's a vector quantity, which means it has both a magnitude and a direction.
When dealing with rotating objects, like a potter's wheel, angular velocity is expressed in radians per second (rad/s). Radians are a unit of angular measure used in many areas of mathematics.

• Initial Angular Velocity (\(\omega_0\)): This is the speed of rotation at the beginning of a time interval.
• Final Angular Velocity (\(\omega\)): This is the speed at the end of the time interval, after the object has undergone some angular acceleration.

Angular velocity connects closely with angular displacement and angular acceleration, as it helps describe how the rotation of an object changes over time.

Angular Acceleration

Angular acceleration describes the rate at which angular velocity changes. It's similar to linear acceleration but applies to rotating objects.
Angular acceleration is expressed in radians per second squared (rad/s²). When an object experiences angular acceleration, it means the speed of its rotation is either increasing or decreasing.
In the case of Emilie's potter's wheel:

• The constant angular acceleration of 2.25 rad/s² means the wheel's angular velocity increases steadily over time.
• This incremental increase results in the wheel spinning faster and covering more angle as time progresses.
• Angular acceleration, like angular velocity, is a vector quantity.

Understanding this concept is essential when solving motion problems involving rotating objects, as it helps determine how the motion evolves.

Angular Displacement

Angular displacement measures how much an object has rotated from its starting point. It is the angle in radians that the object sweeps through during a given time frame.
Unlike distance in linear motion, which is a scalar quantity, angular displacement provides information about both direction and magnitude in the context of rotation.

• In Emilie's exercise, the wheel rotates through an angular displacement of 60 radians over 4 seconds.
• This displacement helps us understand the total extent of rotation regardless of any changes in velocity or acceleration.
• Utilizing equations of motion for rotation can aid in determining other rotational parameters, like initial angular velocity, given angular displacement, time, and acceleration.

Understanding angular displacement is fundamental to grasping how rotations are quantified and analyzing the dynamics of rotational systems.