Problem 9 - University Physics with Modern Physics

Question

A bicycle wheel has an initial angular velocity of 1.50 \(\mathrm{rad} / \mathrm{s}\) . (a) If its angular acceleration is constant and equal to 0.300 \(\mathrm{rad} / \mathrm{s}^{2}\) , what is its angular velocity at \(t=2.50 \mathrm{s} ?\) (b) Through what angle has the wheel turned between \(t=0\) and \(t=2.50 \mathrm{s} ?\)

Step-by-Step Solution

Answer

(a) 2.25 rad/s; (b) 4.6875 radians.

Step 1: Identify Known Variables

First, note the given variables. The initial angular velocity \( \omega_0 \) is 1.50 rad/s and the constant angular acceleration \( \alpha \) is 0.300 rad/s².

Step 2: Use Angular Velocity Formula

To find the angular velocity \( \omega \) at \( t = 2.50 \) s, use the equation \( \omega = \omega_0 + \alpha t \). Substituting the known values gives \( \omega = 1.50 + (0.300)(2.50) \).

Step 3: Calculate Angular Velocity

Perform the calculation: \( \omega = 1.50 + 0.300 \times 2.50 = 1.50 + 0.750 = 2.25 \) rad/s. Thus, the angular velocity at \( t = 2.50 \) s is 2.25 rad/s.

Step 4: Use Angular Displacement Formula

To find the angle turned \( \theta \), use the formula \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \). Substituting the known values gives \( \theta = (1.50)(2.50) + \frac{1}{2}(0.300)(2.50)^2 \).

Step 5: Calculate Angular Displacement

Perform the calculations: \( \theta = 3.75 + \frac{1}{2}(0.300)(6.25) = 3.75 + 0.9375 = 4.6875 \) rad. Thus, the wheel turns through an angle of 4.6875 radians.

Key Concepts

Angular Velocity

Angular velocity is a key concept when dealing with rotational motion. It measures how fast an object rotates around a central point or axis. Unlike linear velocity, which measures how fast an object moves in a straight line, angular velocity is concerned with the rate of rotation.

• Initial Angular Velocity (\( \omega_0 \)): This is the starting point of the rotation. In the exercise, the bicycle wheel begins with an angular velocity of 1.50 rad/s. This means that initially, the wheel rotates 1.50 radians every second.
• Calculation: To find the angular velocity at a later time, we use the formula \( \omega = \omega_0 + \alpha t \), where \( \alpha \) is the angular acceleration, and \( t \) is time. This is the rate at which the wheel's angular speed increases or decreases. For instance, at \( t = 2.50 \) seconds and an angular acceleration of 0.300 rad/s², the wheel's angular velocity becomes 2.25 rad/s.

Angular velocity is vital for understanding how quickly an object turns. It helps in predicting future positions and times by knowing the initial speed of rotation and how that speed changes over time.
Understanding angular velocity offers insight into the wheel's dynamic behavior as it spins.

Angular Acceleration

Angular acceleration describes how quickly the angular velocity of an object changes with time. It is an essential factor in determining how a rotating object speeds up or slows down.

• Constant Angular Acceleration (\( \alpha \)): In the given exercise, the wheel's angular acceleration is 0.300 rad/s². This means that every second, the angular velocity of the wheel increases by 0.300 radians per second.
• Effect on Angular Velocity: Angular acceleration is used in the formula \( \omega = \omega_0 + \alpha t \) to calculate the final angular velocity. The constant angular acceleration ensures a predictable change in velocity over time.

Angular acceleration is crucial because it reflects how rapidly an object can change its state of rotation. It helps in determining the time it would take for a spinning object to reach a particular speed, or conversely, how long it might take for it to stop.

Angular Displacement

Angular displacement refers to the angle through which a point or line has been rotated in a specified sense about a specified axis. It is the angle between the initial and final positions of a rotating body.

• Formula for Angular Displacement (\( \theta \)): In the scenario, we use \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \) to find out how much the bicycle wheel has turned. This formula combines the initial angular velocity and the effect of angular acceleration over time to calculate total displacement.
• Calculated Example: With an initial angular velocity of 1.50 rad/s and angular acceleration of 0.300 rad/s², after 2.50 seconds, the wheel has turned through 4.6875 radians. This reveals the cumulative effect of both its starting speed and steady acceleration on position.

Understanding angular displacement helps visualize the entire extent an object has rotated over time. It's a fundamental part of rotational dynamics as it connects with both angular velocity and angular acceleration to describe a complete picture of rotational movement.