A wheel, starting from rest, has a constant angular acceleration of 1.8 rad/s^2. In a 3.1-s interval, it turns through an angle of 142 rad. How long has the wheel been in motion at the start of this 3.1-s interval?
To find out how long the wheel has been in motion at the start of the 3.1-second interval, we need to determine the initial angular velocity of the wheel.
We know that the wheel starts from rest, so the initial angular velocity is zero radians per second. The angular acceleration is given as 1.8 rad/s^2, and the angle turned through in the 3.1-second interval is 142 rad.
To find the time it takes for the wheel to reach this angle, we can use the following equation:
θ = ω_i * t + (1/2) * α * t^2
Where θ is the angle turned through, ω_i is the initial angular velocity, α is the angular acceleration, and t is the time.
Since the initial angular velocity is zero, the equation simplifies to:
θ = (1/2) * α * t^2
Rearranging the equation to solve for t:
t = √(2θ / α)
Plugging in the given values, we have:
t = √(2 * 142 rad / 1.8 rad/s^2)
t = √(284 / 1.8)
t = √(157.78)
t ≈ 12.56 s
Therefore, the wheel has been in motion for approximately 12.56 seconds at the start of the 3.1-second interval.