A wheel of diameter 40.0 cm starts from rest and rotates with a constant angular acceleration of 3.00 rad/s^2. Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship (a) a_rad = ω^2r and (b) a_rad = v^2/r

Verified step by step guidance

1. Calculate the angular velocity (ω) at the instant the wheel completes its second revolution. Use the formula ω = sqrt(ω₀² + 2αθ), where ω₀ is the initial angular velocity (0 rad/s since it starts from rest), α is the angular acceleration (3.00 rad/s²), and θ is the angular displacement (4π rad for two revolutions).
2. Compute the radius (r) of the wheel from its diameter. Since the diameter is 40.0 cm, the radius r = diameter / 2 = 20.0 cm. Convert this to meters by dividing by 100, so r = 0.20 m.
3. Use the relationship a_rad = ω²r to find the radial acceleration. Substitute the values of ω and r obtained from the previous steps.
4. Alternatively, calculate the linear velocity (v) of a point on the rim using the formula v = ωr. Use the angular velocity (ω) and radius (r) from the previous steps.
5. Use the second relationship a_rad = v²/r to find the radial acceleration. Substitute the values of v and r obtained from the previous steps.

Key Concepts

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically measured in radians per second squared (rad/s²). In this scenario, the wheel has a constant angular acceleration of 3.00 rad/s², which means its angular velocity increases steadily as it rotates. Understanding angular acceleration is crucial for determining how quickly the wheel speeds up and affects the linear velocity of points on its rim.

Radial (Centripetal) Acceleration

Radial acceleration, also known as centripetal acceleration, is the acceleration directed towards the center of a circular path, necessary for an object to maintain circular motion. It can be calculated using the formula a_rad = ω²r, where ω is the angular velocity and r is the radius of the circular path. This concept is essential for analyzing the forces acting on a point on the rim of the wheel as it rotates.

Relationship Between Linear and Angular Quantities

The relationship between linear and angular quantities is fundamental in rotational motion. Linear velocity (v) is related to angular velocity (ω) by the equation v = ωr, where r is the radius. This relationship allows us to convert between linear and angular measurements, enabling the use of different formulas, such as a_rad = v²/r, to calculate radial acceleration based on linear velocity.