A wheel of diameter 40.0 cm starts from rest and rotates with a constant angular acceleration of 3.00 rad/s². Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship a_rad = ω²r.

Verified step by step guidance

1. Determine the angular displacement (θ) for two revolutions. Since one revolution corresponds to an angular displacement of 2π radians, two revolutions correspond to θ = 4π radians.
2. Use the kinematic equation for rotational motion to find the angular velocity (ω) at the instant the wheel completes its second revolution. The equation is: θ = ω₀t + (1/2)αt². Since the wheel starts from rest (ω₀ = 0), simplify to θ = (1/2)αt². Solve for time (t) first, then use ω = ω₀ + αt to find ω.
3. For part (a), calculate the radial acceleration using the formula a_rad = ω²r. Here, r is the radius of the wheel, which is half the diameter (r = 0.40 m / 2). Substitute the value of ω obtained in the previous step and the radius into the formula.
4. For part (b), calculate the radial acceleration using the formula a_rad = v²/r. First, find the linear velocity (v) using the relationship v = ωr, where ω is the angular velocity and r is the radius. Then substitute v and r into the formula a_rad = v²/r.
5. Ensure all units are consistent (e.g., convert the diameter to meters) and verify the calculations for both parts (a) and (b) to confirm the radial acceleration values are consistent with the given angular acceleration and rotational motion.

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 calculating the angular velocity at any given time, which is necessary for determining radial acceleration.

Radial Acceleration

Radial acceleration, also known as centripetal acceleration, is the acceleration directed towards the center of a circular path. It can be calculated using the formulas a_rad = ω²r or a_rad = v²/r, where ω is the angular velocity, v is the linear velocity, and r is the radius of the circular path. This concept is essential for understanding how objects moving in a circle experience acceleration that keeps them in that circular motion.

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 of the circular path. This relationship allows us to convert between linear and angular measurements, which is necessary for calculating radial acceleration when given angular parameters, as in the case of the rotating wheel.