A dentist's drill starts from rest. After 3.62 s of constant angular acceleration it turns at a rate of 2.70E+4 revolutions per minute. Find the drill's angular acceleration.
Determine the angle through which the drill rotates during this period.
To find the angular acceleration of the drill, we can use the formula:
angular acceleration (α) = (final angular velocity ( ω ) - initial angular velocity (ω0)) / time (t)
Given:
Initial angular velocity ( ω0) = 0 (as the drill starts from rest)
Final angular velocity ( ω ) = 2.70E+4 revolutions per minute
Time (t) = 3.62 s
First, we need to convert the final angular velocity from revolutions per minute to radians per second. Since 1 revolution = 2π radians, we can use the conversion factor:
1 revolution/minute = 2π radians/60 seconds
So, the final angular velocity in radians per second is:
ω = (2.70E+4 rev/min) * (2π rad/1 rev) * (1 min/60 s) = 2827.4334 radians/sec
Now we can substitute the values into the formula to find the angular acceleration:
α = (2827.4334 rad/sec - 0 rad/sec) / 3.62 sec = 781.9685 rad/sec^2
Therefore, the drill's angular acceleration is 781.9685 rad/sec^2.
To determine the angle through which the drill rotates during this period, we can use the kinematic equation for rotational motion:
Δθ = ω0t + 1/2αt^2
Since the initial angular velocity is 0, the equation simplifies to:
Δθ = 1/2αt^2
Substituting the values, we have:
Δθ = 1/2(781.9685 rad/sec^2)(3.62 s)^2 = 5146.4802 radians
Therefore, the drill rotates through an angle of 5146.4802 radians during this period.