Find the angular velocity of a body rotating with an acceleration of 2 rev/s square as it completes the 5th revolution after the start.
I assume they want the angular velocity in radians/sec units.
After 5 revolutions, the angle turned is
A = 10 pi radians
The angular acceleration rate is
alpha = 4 pi radians/s^2
The time since starting is given by
A = (1/2)*alpha*t^2
t^2 = 2A/alpha = 5 sec^2
t = 2.236 s
w(at t = 2.236 s) = alpha*t = 28.1 rad/s
To find the angular velocity of a body rotating with an acceleration, you need to use the equations of rotational motion. The equation that relates the angular acceleration (α), angular velocity (ω), and time (t) is given by:
ω = ω0 + αt
Where:
ω0 is the initial angular velocity (which is usually assumed to be zero if not given)
α is the angular acceleration
t is the time
In this case, the angular acceleration is given as 2 rev/s², and we want to find the angular velocity after the body completes the 5th revolution. Let's break down the solution into steps:
Step 1: Convert revolutions to radians
Since angular velocity is usually measured in radians per second, we need to convert the 5 revolutions into radians. Remember that 1 revolution is equivalent to 2π radians. Therefore, 5 revolutions would be 5 * 2π radians.
Step 2: Determine the time taken
Given that the body completes the 5th revolution from the start, we need to find the time it took to reach that point. To do this, we can use the formula for the number of revolutions completed (n) in terms of time:
n = ω0t + (1/2)αt²
Since ω0 is assumed to be zero in this case, the equation reduces to:
n = (1/2)αt²
Substituting the given values, we have:
5 = (1/2)(2)t²
Step 3: Solve for time
Simplify and solve the equation for t:
10t² = 5
t² = 5/10
t = √(1/2)
Step 4: Calculate the angular velocity
Now that we have the time (t), we can substitute it into the equation for angular velocity:
ω = ω0 + αt
Since ω0 is assumed to be zero, the equation further simplifies to:
ω = αt
Substituting the given values, we have:
ω = 2 rev/s² * √(1/2) s
Therefore, the angular velocity of the body, after completing the 5th revolution from the start, is 2√(1/2) rev/s².