A dentist's drill starts from rest. After 3.00 s of constant angular acceleration it turns at a rate of 2.8 104 rev/min.
(a) Find the drill's angular acceleration.
(b) Determine the angle (in radians) through which the drill rotates during this period.
155.5
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 that the initial angular velocity (ω₀) is 0 rev/min, the final angular velocity (ω) is 2.8 x 10^4 rev/min, and the time (t) is 3.00 s, we can plug in these values into the formula to find the angular acceleration.
Step 1: Convert angular velocity from rev/min to rad/s:
We know that 1 rev = 2π rad, and 1 min = 60 s.
So, to convert rev/min to rad/s, we need to multiply by (2π rad/1 rev) * (1 min/60 s):
final angular velocity (ω) = (2.8 x 10^4 rev/min) * (2π rad/1 rev) * (1 min/60 s)
= (2.8 x 10^4 * 2π) * (1/60) rad/s
Let's calculate this value:
final angular velocity (ω) ≈ (2.8 x 10^4 * 2π) * (1/60) rad/s
≈ 2938.53 rad/s
Step 2: Substitute the values into the formula and calculate the angular acceleration:
angular acceleration (α) = (final angular velocity (ω) - initial angular velocity (ω₀)) / time (t)
= (2938.53 rad/s - 0 rad/s) / 3.00 s
≈ 979.51 rad/s²
Therefore, the drill's angular acceleration is approximately 979.51 rad/s².
To determine the angle (in radians) through which the drill rotates during this period, we can use the formula:
θ = ω₀t + 0.5αt²
Given that the initial angular velocity (ω₀) is 0 rad/s, the time (t) is 3.00 s, and the angular acceleration (α) is 979.51 rad/s², we can substitute these values into the formula to find the angle (θ).
θ = (0 rad/s)(3.00 s) + 0.5(979.51 rad/s²)(3.00 s)²
Let's calculate this value:
θ ≈ 0 rad + 0.5(979.51 rad/s²)(9.00 s²)
≈ 0 rad + 4407.80 rad
≈ 4407.80 rad
Therefore, the angle through which the drill rotates during this period is approximately 4407.80 radians.