A wheel starting from rest is uniformly accelerated with alpha=4 rad/seconds square for 10 seconds. It is then allowed to rotate uniformly for the next two seconds and is finally brought to rest in the next 10 seconds. Find the total angle rotated by the wheel
1/2*400+180
To find the total angle rotated by the wheel, we need to consider the three different phases of motion - the acceleration phase, the constant velocity phase, and the deceleration phase.
Acceleration Phase:
In this phase, the wheel starts from rest and undergoes uniform acceleration. We are given that the angular acceleration (alpha) is 4 rad/s^2 and the time (t) is 10 seconds.
To find the angle rotated (theta) during this phase, we can use the formula:
theta = 0.5 * alpha * t^2
Plugging in the values, we get:
theta = 0.5 * 4 * (10^2) = 200 rad
Constant Velocity Phase:
During this phase, the wheel rotates uniformly for 2 seconds. Since the velocity is constant, we can use the formula:
theta = w * t
where w is the angular velocity. Since the angular velocity is constant, we can use the value from the previous phase, where it was at the end of the acceleration phase, which is 4 * 10 = 40 rad/s.
Plugging in the values, we get:
theta = 40 * 2 = 80 rad
Deceleration Phase:
During this phase, the wheel is brought to rest in 10 seconds. The final angular velocity is zero, and the initial angular velocity can be found using the formula:
w = w0 + alpha * t
Since the final angular velocity (w) is zero and the time (t) is 10 seconds, we can rearrange the formula to solve for the initial angular velocity (w0):
w0 = -alpha * t
Plugging in the values, we get:
w0 = -4 * 10 = -40 rad/s
Using the formula for the angle, we get:
theta = w * t + 0.5 * alpha * t^2
Plugging in the values, we get:
theta = (-40 * 10) + 0.5 * 4 * (10^2) = -400 + 200 = -200 rad
Note: The negative sign indicates that the wheel is rotating in the opposite direction during the deceleration phase.
Total Angle Rotated:
To find the total angle rotated, we sum up the angles from each phase:
Total angle = theta (acceleration phase) + theta (constant velocity phase) + theta (deceleration phase)
= 200 rad + 80 rad + (-200 rad)
= 80 rad
Therefore, the total angle rotated by the wheel is 80 radians.
displacement=1/2 a*10^2+a*10*2+1/2 a*10*10
check that.