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 (b) a_rad = v^2/r

Verified step by step guidance

1. Determine the radius of the wheel by dividing the diameter by 2. Since the diameter is 40.0 cm, the radius (r) is 20.0 cm. Convert this radius into meters by dividing by 100, as 1 m = 100 cm.
2. Calculate the angular displacement (\(\theta\)) after completing two revolutions. Since one revolution is \(2\pi\) radians, two revolutions will be \(2 \times 2\pi = 4\pi\) radians.
3. Use the formula for angular displacement \(\theta = \omega_0 t + \frac{1}{2} \alpha t^2\) where \(\omega_0\) is the initial angular velocity (0 rad/s, as it starts from rest), \(\alpha\) is the angular acceleration, and t is the time. Solve for t.
4. Calculate the tangential velocity (v) at the point on the rim using the relationship \(v = r\omega\), where \(\omega\) is the angular velocity at time t, calculated using \(\omega = \omega_0 + \alpha t\).
5. Compute the radial acceleration (\(a_{rad}\)) using the formula \(a_{rad} = \frac{v^2}{r}\). Substitute the values of v and r obtained in the previous steps to find \(a_{rad}\).

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, meaning its rotational speed increases steadily as it spins. This concept is crucial for determining the angular velocity at any given time during the wheel's motion.

Tangential Velocity

Tangential velocity refers to the linear speed of a point on the circumference of a rotating object, calculated as the product of the angular velocity and the radius of the rotation. As the wheel accelerates, the tangential velocity increases, which can be derived from the angular acceleration and the time elapsed. This velocity is essential for calculating the radial acceleration at any point in the wheel's rotation.

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 its circular motion. It is calculated using the formula a_rad = v²/r, where v is the tangential velocity and r is the radius of the circular path. Understanding this concept is vital for determining the acceleration experienced by a point on the rim of the wheel as it completes its revolutions.