A 6.0-cm-diameter gear rotates with angular velocity ω = ( 20 ─ ½ t² ) rad/s where t is in seconds. At t = 4.0 s, what are: b. The tangential acceleration of a tooth on the gear?

Verified step by step guidance

1. Identify the expression for angular velocity, \( \omega = 20 - \frac{1}{2}t^2 \) rad/s, and note that the time given is t = 4.0 s.
2. Calculate the angular acceleration, \( \alpha \), by differentiating the angular velocity equation with respect to time, t. The derivative of \( \omega \) with respect to t gives \( \alpha = -t \).
3. Substitute t = 4.0 s into the angular acceleration equation to find \( \alpha \) at that specific time.
4. Use the formula for tangential acceleration, \( a_t = r \alpha \), where \( r \) is the radius of the gear. Convert the diameter of the gear to radius by dividing by 2, so \( r = 3.0 \) cm or \( 0.03 \) m.
5. Substitute the values of \( r \) and \( \alpha \) into the tangential acceleration formula to find the tangential acceleration of a tooth on the gear at t = 4.0 s.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second. In this question, the angular velocity ω is given as a function of time, indicating that it changes as time progresses. Understanding how to differentiate this function is crucial for finding the angular acceleration and subsequently the tangential acceleration.

Tangential Acceleration

Tangential acceleration refers to the rate of change of the linear velocity of a point on a rotating object. It can be calculated using the formula a_t = r * α, where r is the radius and α is the angular acceleration. This concept is essential for determining how quickly a point on the gear is speeding up or slowing down as it rotates.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically expressed in radians per second squared. It can be derived by differentiating the angular velocity function with respect to time. In this problem, calculating angular acceleration is necessary to find the tangential acceleration of a tooth on the gear at a specific time.