Final answer:
It takes approximately 5.69 seconds for a wheel to increase its rotational speed from 24.0 rpm to 62.0 rpm with an angular acceleration of 0.7 rad/s² after converting the speeds to rad/s and using the formula for angular motion.
Step-by-step explanation:
The student is asking how long it takes for a wheel to increase its rotational speed from 24.0 revolutions per minute (rpm) to 62.0 rpm given an angular acceleration of 0.7 radians/second2. To calculate this, we need to convert the given revolutions per minute into radians per second, and then use the formula that relates angular acceleration (alpha, α) with change in angular velocity (δω) and time (t), which is δω = α * t.
First, converting the initial and final angular velocities:
- Initial angular velocity (ωi) = (24.0 rpm * 2π rad/rev) / 60 s/min = 2.51 rad/s
- Final angular velocity (ωf) = (62.0 rpm * 2π rad/rev) / 60 s/min = 6.49 rad/s
Using the equation, we rearrange to solve for time (t):
- t = (ωf - ωi) / α = (6.49 rad/s - 2.51 rad/s) / 0.7 rad/s2 ≈ 5.69 s
Therefore, it takes approximately 5.69 seconds for the wheel to increase from 24.0 rpm to 62.0 rpm at an angular acceleration of 0.7 rad/s2.