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 = v²/r.

Verified step by step guidance

1. Step 1: Start by determining 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. Step 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: $\omega ^2\; =\; \omega \_0^2\; +\; 2\alpha \theta$, where ω_0 is the initial angular velocity (0 rad/s), α is the angular acceleration (3.00 rad/s²), and θ is the angular displacement (4π radians).
3. Step 3: Calculate the linear velocity (v) of a point on the rim of the wheel using the relationship between linear and angular velocity: $v\; =\; r\omega$, where r is the radius of the wheel. The radius can be found from the diameter: r = 40.0 cm / 2 = 20.0 cm = 0.200 m.
4. Step 4: Substitute the value of v into the formula for radial acceleration: $a\_\{rad\}\; =\; \backslash (\backslash frac\{v^2\}\{r\}\backslash )$. Use the radius (r = 0.200 m) and the linear velocity (v) calculated in the previous step.
5. Step 5: Simplify the expression to find the radial acceleration (a_rad). Ensure all units are consistent (meters, seconds, etc.) throughout the calculation.

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 uniformly as it rotates. This concept is crucial for determining the wheel's angular velocity at any given time.

Tangential Velocity

Tangential velocity is the linear speed of a point on the circumference of a rotating object, calculated as the product of the radius and the angular velocity (v = rω). As the wheel rotates, its tangential velocity increases due to the constant angular acceleration. This velocity is essential for calculating the radial acceleration at a specific point on the wheel.

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 can be 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 solving the problem regarding the acceleration of a point on the wheel's rim.