An electric motor is attached to a small wheel that is rotating at a rate of 150 rev \min . if the motor is switched off so that it has a negative angular acceleration of 2.2 rad\s^2 , how long dose it take the wheel to stop? And how many revolutions has the wheel turned during this time interval ?
First, we need to convert the initial rotation rate to radians per second:
150 rev/min * 2π rad/rev * 1/60 min/s = 15.7 rad/s
Next, we can use the formula for angular acceleration to find the time it takes for the wheel to stop:
ωf = ωi + αt
0 = 15.7 rad/s - 2.2 rad/s^2 * t
t = 7.14 s
Finally, we can use the formula for angular displacement to find how many revolutions the wheel has turned during this time interval:
θ = ωit + 1/2 αt^2
θ = 15.7 rad/s * 7.14 s + 1/2 (-2.2 rad/s^2) * (7.14 s)^2
θ = 55.9 rad
55.9 rad * 1 rev/2π rad = 8.9 rev (rounded to one decimal place)
Therefore, the wheel takes 7.14 seconds to stop and has turned 8.9 revolutions during that time interval.
To find the time it takes for the wheel to stop, we can use the equation of motion for rotational motion:
angular acceleration (α) = final angular velocity (ω) - initial angular velocity (ω₀) / time (t)
Given:
Initial angular velocity, ω₀ = 150 rev/min
Final angular velocity, ω = 0 (since the wheel stops)
Angular acceleration, α = -2.2 rad/s²
Converting the initial angular velocity to rad/s:
Initial angular velocity, ω₀ = (150 rev/min) * (2π rad/rev) * (1 min/60 s)
ω₀ ≈ 15.708 rad/s
Using the equation of motion, we have:
-2.2 = 0 - 15.708 / t
Simplifying the equation, we get:
-2.2t = -15.708
Divide both sides by -2.2 to solve for t:
t ≈ -15.708 / -2.2
t ≈ 7.13 s
Therefore, it takes approximately 7.13 seconds for the wheel to stop.
To find the number of revolutions the wheel has turned during this time interval, we can use the formula:
Number of revolutions = (Final angular velocity - Initial angular velocity) / (2π)
The final angular velocity is 0 rad/s, and the initial angular velocity is 15.708 rad/s. Plugging these values into the formula, we have:
Number of revolutions = (0 - 15.708) / (2π)
Number of revolutions ≈ -7.925 rev
Since the number of revolutions cannot be negative, we take the absolute value:
Number of revolutions ≈ 7.925 rev
Therefore, the wheel has turned approximately 7.925 revolutions during this time interval.