A bicycle wheel has an initial angular velocity of 1.50 rad/s. (a) If its angular acceleration is constant and equal to 0.200 rad/s2, what is its angular velocity at t = 2.50 s? (b) Through what angle has the wheel turned between t = 0 and t = 2.50 s?

Ch 09: Rotation of Rigid Bodies

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 09: Rotation of Rigid Bodies

문제 9a

9장, 문제 9a

A bicycle wheel has an initial angular velocity of 1.50 rad/s. (a) If its angular acceleration is constant and equal to 0.200 rad/s², what is its angular velocity at t = 2.50 s? (b) Through what angle has the wheel turned between t = 0 and t = 2.50 s?

검증된 단계별 안내

Step 1: Identify the given values for part (a). The initial angular velocity \( \omega_0 \) is 1.50 rad/s, the angular acceleration \( \alpha \) is 0.200 rad/s^2, and the time \( t \) is 2.50 s. Use the kinematic equation for angular velocity: \( \omega = \omega_0 + \alpha t \).

Step 2: Substitute the given values into the equation \( \omega = \omega_0 + \alpha t \). This will allow you to calculate the angular velocity \( \omega \) at \( t = 2.50 \) s.

Step 3: For part (b), identify the given values again: \( \omega_0 = 1.50 \) rad/s, \( \alpha = 0.200 \) rad/s^2, and \( t = 2.50 \) s. Use the kinematic equation for angular displacement: \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \).

Step 4: Substitute the given values into the equation \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \). This will allow you to calculate the angular displacement \( \theta \), which represents the angle the wheel has turned.

Step 5: Ensure that the units are consistent throughout the calculations (radians for angular quantities, seconds for time) and verify the results for both angular velocity and angular displacement using the respective equations.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second (rad/s). It indicates the rate of change of the angular position of the object. In this question, the initial angular velocity of the bicycle wheel is given as 1.50 rad/s, which serves as the starting point for calculating its angular velocity after a certain time.

Angular Acceleration

Angular acceleration refers to the rate of change of angular velocity over time, measured in radians per second squared (rad/s²). It indicates how quickly the angular velocity of an object is increasing or decreasing. In this scenario, the constant angular acceleration of 0.200 rad/s² affects the wheel's angular velocity over the specified time interval.

Angular Displacement

Angular displacement is the angle through which an object has rotated about a specific axis, measured in radians. It can be calculated using the initial angular velocity, angular acceleration, and time. In this problem, determining the angle through which the bicycle wheel has turned involves applying the equations of motion for rotational dynamics, which relate these quantities.