A variable-speed drill, initially turning at 500 rpm, speeds up to 1300 rpm in a time interval of 0.8 s. What is its average rotational acceleration?
This is just like linear acceleration, but going around. It's just a speed, velocity is measured in rotations/sec instead of meters/sec.
So, since the speed of the drill changed from 500-1300rpm, the change was 800rpm = 800/60 = 13.33 rot/sec.
That took 0.8s, so the average acceleration is
(13.33rot/s)/(0.8s) = 16.66 rot/s^2
To find the average rotational acceleration, we need to use the formula:
Average rotational acceleration (α) = (Change in rotational speed (Δω)) / (Time interval (Δt))
Given:
Initial rotational speed (ω1) = 500 rpm
Final rotational speed (ω2) = 1300 rpm
Time interval (Δt) = 0.8 s
Step 1: Convert the rotational speeds from rpm to rad/s
RPM stands for revolutions per minute, and since we need the rotational speeds in rad/s, we need to convert them.
1 revolution = 2π radians
Initial rotational speed (ω1) = 500 rpm
ω1 = 500 rpm * 2π rad/1 min * 1 min/60 s
ω1 = 500 * 2π / 60 rad/s
ω1 ≈ 52.36 rad/s
Final rotational speed (ω2) = 1300 rpm
ω2 = 1300 rpm * 2π rad/1 min * 1 min/60 s
ω2 = 1300 * 2π / 60 rad/s
ω2 ≈ 136.65 rad/s
Step 2: Calculate the change in rotational speed (Δω)
Δω = ω2 - ω1
Δω = 136.65 rad/s - 52.36 rad/s
Δω ≈ 84.29 rad/s
Step 3: Calculate the average rotational acceleration (α)
α = Δω / Δt
α = 84.29 rad/s / 0.8 s
α ≈ 105.36 rad/s²
Therefore, the average rotational acceleration of the variable-speed drill is approximately 105.36 rad/s².
To find the average rotational acceleration of the variable-speed drill, we can use the equation:
Average rotational acceleration = (Final rotational speed - Initial rotational speed) / Time interval
Given:
Initial rotational speed (ω1) = 500 rpm
Final rotational speed (ω2) = 1300 rpm
Time interval (Δt) = 0.8 s
First, we need to convert the rotational speeds from rpm to radians per second (rad/s) because the SI unit for rotational speed is radians per second.
1 revolution = 2π radians
1 minute = 60 seconds
So, to convert from rpm to rad/s, we can use the conversion factor:
1 rpm = (2π/60) rad/s
Initial rotational speed in rad/s (ω1) = 500 rpm * (2π/60) rad/s = (500π/60) rad/s
Final rotational speed in rad/s (ω2) = 1300 rpm * (2π/60) rad/s = (1300π/60) rad/s
Now we can use the equation to compute the average rotational acceleration:
Average rotational acceleration = ((Final rotational speed) - (Initial rotational speed)) / (Time interval)
= ((1300π/60) rad/s - (500π/60) rad/s) / (0.8 s)
= ((1300π - 500π) rad/s) / (60 * 0.8) s
= (800π rad/s) / (48) s
≈ 52.36 rad/s²
So, the average rotational acceleration of the variable-speed drill is approximately 52.36 rad/s².