A dentist's drill starts from rest. After 2.50 s of constant angular acceleration it turns at a rate of 3.0 104 rev/min.
(a) Find the drill's angular acceleration.
rad/s2
(b) Determine the angle (in radians) through which the drill rotates during this period.
rad
(a) Convert 30,000 rpm to radians per second. Call it w.
w = 52.36 rad/s
Angular acceleration = w/2.5 rad/s^2
(b) Theta = (average angular velocity)*time = (w/2)*2.5 radians.
To find the angular acceleration of the dentist's drill, we can use the formula:
Angular acceleration (α) = (Final angular velocity (ω) - Initial angular velocity (ω₀)) / Time (t)
Given:
Final angular velocity (ω) = 3.0 × 10^4 rev/min
Time (t) = 2.50 s
First, let's convert the final angular velocity from rev/min to rad/s. Since 1 rev = 2π rad, we have:
Final angular velocity (ω) = 3.0 × 10^4 rev/min * (2π rad/1 rev) * (1 min / 60 s)
= 3.0 × 10^4 * 2π / 60 rad/s
Now we can substitute the values into the formula to find the angular acceleration:
Angular acceleration (α) = (3.0 × 10^4 * 2π / 60 rad/s - 0 rad/s) / 2.50 s
Simplifying the equation gives us the angular acceleration in rad/s².
To determine the angle (in radians) through which the drill rotates during this period, we can use the formula:
Angular displacement (θ) = (Initial angular velocity (ω₀) * Time (t)) + (0.5 * Angular acceleration (α) * Time (t)^2)
We already know the initial angular velocity (ω₀) is 0 rad/s, the time (t) is 2.50 s, and we calculated the angular acceleration (α) in part (a). Substituting these values into the formula will give us the angle in radians.