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: a. The gear's angular acceleration?

Verified step by step guidance

1. Identify the expression for angular velocity, \( \omega = 20 - \frac{1}{2}t^2 \) rad/s.
2. To find the angular acceleration, differentiate the angular velocity expression with respect to time, t. This gives \( \alpha = \frac{d\omega}{dt} \).
3. Differentiate the given expression \( \omega = 20 - \frac{1}{2}t^2 \) with respect to t. Use the power rule for differentiation, where the derivative of \( t^n \) is \( n \cdot t^{n-1} \).
4. Substitute t = 4.0 s into the differentiated expression to find the angular acceleration at that specific time.
5. The result from the substitution will give you the angular acceleration in rad/s^2 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 (rad/s). It indicates the rate of change of angular displacement over time. In this question, the angular velocity is given as a function of time, which means it changes as time progresses.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically measured in radians per second squared (rad/s²). It can be calculated by taking the derivative of the angular velocity function with respect to time. In this case, finding the angular acceleration at a specific time involves differentiating the provided angular velocity equation.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a quantity changes. In the context of this problem, differentiation is used to determine angular acceleration by calculating the derivative of the angular velocity function with respect to time. This process allows us to analyze how the gear's rotation speed changes over time.