A remote-controlled car’s wheel accelerates at 22.6 rad/s2.
If the wheel begins with an angular speed of 10.6 rad/s, what is the wheel’s angular speed after exactly three full turns?
d=wi*t+alpha*t^2 /2
3*2PI=10.6*t+1/2 22.6*t^2
solve for time t.
wf=wi+22.6*t solve for wf
To find the wheel's angular speed after exactly three full turns, we need to consider the given acceleration and initial angular speed.
The equation for angular acceleration is:
angular acceleration = (final angular speed - initial angular speed) / time
Since the wheel is accelerating at a constant rate, the average angular speed over the three turns is the average of the initial and final angular speeds. Therefore, the equation becomes:
angular acceleration = (average angular speed - initial angular speed) / time
Since the wheel is accelerating, the final angular speed will be greater than the initial angular speed.
In this case, the initial angular speed is 10.6 rad/s, and the angular acceleration is 22.6 rad/s^2. We want to find the final angular speed after three full turns.
First, we need to determine the time it takes to complete three full turns. Since one full turn is equal to 2π radians, three full turns are equal to 3 * 2π = 6π radians.
We know the angular acceleration and initial angular speed, so let's find the average angular speed using the equation mentioned earlier:
angular acceleration = (average angular speed - initial angular speed) / time
Rearranging the equation to find the average angular speed:
average angular speed = angular acceleration * time + initial angular speed
Plugging in the values:
average angular speed = 22.6 rad/s^2 * t + 10.6 rad/s
Now, we need to find the time it takes to complete three full turns. To do this, we can use the formula for angular displacement and rearrange it to solve for time:
angular displacement = (average angular speed + initial angular speed) / 2 * time
Substituting the values:
6π = (22.6 rad/s^2 * t + 10.6 rad/s + 10.6 rad/s) / 2 * t
Simplifying:
6π = (22.6 rad/s^2 * t + 21.2 rad/s) / 2 * t
Rearranging the equation to solve for t:
2 * t * 6π = 22.6 rad/s^2 * t + 21.2 rad/s
12πt = 22.6 rad/s^2 * t + 21.2 rad/s
Moving the terms with t to one side:
12πt - 22.6 rad/s^2 * t = 21.2 rad/s
t(12π - 22.6 rad/s^2) = 21.2 rad/s
Finally, solving for t:
t = 21.2 rad/s / (12π - 22.6 rad/s^2)
Once we have the value of t, we can find the average angular speed by substituting it back into the equation:
average angular speed = 22.6 rad/s^2 * t + 10.6 rad/s
After finding the average angular speed, we can add it to the initial angular speed to obtain the final angular speed after three full turns.