Respuesta :
Answer:
b. 81 rad
Explanation:
First of all, we need to understand how much time it takes for the wheel to stop. This can be calculated by using the equation
[tex]\omega(t) = \omega_0 + \alpha t[/tex]
where
[tex]\omega(t)[/tex] is the angular velocity at time t
[tex]\omega_0 = 18 rad/s[/tex] is the initial angular velocity
[tex]\alpha = -2.0 rad/s^2[/tex] is the angular acceleration (negative because the wheel is slowing down)
[tex]t[/tex] is the time
If we substitute [tex]\omega(t)=0[/tex], we can calculate the time t at which the wheel stops:
[tex]0=\omega_0 + \alpha t\\t=\frac{-\omega_0}{\alpha}=-\frac{18 rad/s}{-2.0 rad/s^2}=9 s[/tex]
And now we can use the equation for the angular displacement to calculate the total angle covered by the wheel during this time:
[tex]\theta(t) = \omega_0 t + \frac{1}{2}\alpha t^2 = (18.0 rad/s)(9 s)+\frac{1}{2}(-2.0 rad/s^2)(9 s)^2=81 rad[/tex]
The time when the wheel has to stops, it will have turned through with the angle of 81 rad.
How to find angular speed of an object?
The angular speed or velocity of a body is the rate by which the body changed its angle with respect to the given time.
As the angular acceleration is rate of change of angular speed with time. Thus
[tex]a=\dfrac{\Delta\omega}{t}\\a=\dfrac{\omega-\omega_o}{t}\\\omega=\omega_o}+a{t}\\[/tex]
Here (ω₀) is the initial angular velocity, (a) is the acceleration and (t) is the time.
The wheel initially has an angular velocity of 18 rad/s, and then slow down with the rate of 2.0 rad/s2 which is deceleration of the body.
Let the angular velocity zero, to find out the value of time. Plug in the values in the above formula as,
[tex]0=18-(-2)(t)[/tex]
Here, sign of acceleration is taken negative due to decreasing acceleration. Solving it further,
[tex]t=9\rm \;s[/tex]
By the equation of motion, the angle can be given as,
[tex]\theta=\omega_ot+\dfrac{1}{2}at^2\\\theta=18(9)+(0.5)(-2)(9)^2\\\theta=81\rm \; rad[/tex]
Thus, the time when the wheel has to stops, it will have turned through with the angle of 81 rad.
Learn more about the angular velocity here;
https://brainly.com/question/1452612
