An EF 151 GTA sits on the EF stool and rotates CCW with an initial speed of 60 rpm. After 28 revolutions he stops. Assuming constant acceleration, how long was he rotating?
To solve this problem, we need to first convert the initial speed from rpm to rad/s. Then, we can use the equations of rotational motion to find the time it took for the EF 151 GTA to stop rotating.
Step 1: Convert initial speed from rpm to rad/s.
The conversion factor is 1 revolution = 2π radians.
So, the initial speed in rad/s is:
60 rpm * (2π radians / 1 minute) = 120π rad/min.
Step 2: Convert the initial speed to rad/s.
There are 60 seconds in a minute, so:
120π rad/min * (1 min / 60 s) = 2π rad/s.
Step 3: Use the equations of rotational motion to find the time.
The equation we can use is:
ω_f = ω_i + αt,
where:
ω_f = final angular velocity (0 rad/s since the EF 151 GTA stops),
ω_i = initial angular velocity (2π rad/s),
α = angular acceleration (constant),
t = time.
Since the final angular velocity is 0 rad/s, we can rearrange the equation to solve for time:
0 = 2π rad/s + αt.
Given that the EF 151 GTA rotates CCW, the angular acceleration (α) is negative. So, we have:
0 = 2π rad/s - αt.
The EF 151 GTA stops after 28 revolutions. Since 1 revolution = 2π radians, the angle covered is:
28 revolutions * 2π radians/revolution = 56π radians.
We can now substitute these values into the equation:
0 = 2π rad/s - αt,
56π radians = 2π rad/s * t.
By comparing the coefficients, we can see that α = 2π rad/s^2.
Substituting this value into the equation, we have:
56π radians = 2π rad/s * t.
We can now solve for time:
t = 56π radians / (2π rad/s) = 28 s.
Therefore, the EF 151 GTA rotated for 28 seconds.