Answer:
s = 15.84m
Explanation:
In order to calculate the distance traveled by the bucket, you first use the formula for the torque exerted on the pulley by the weight of the bucket:
[tex]\tau=I\alpha[/tex] (1)
I: moment of inertia of the pulley
α: angular acceleration of the pulley
You can calculate the angular acceleration by taking into account that the torque is also:
[tex]\tau=Wr[/tex] (2)
W: weight of the bucket = Mg = (1.9kg)(9,8m/s^2) = 18.62N
r: radius of the pulley = 0.561m
[tex]\tau=(18.62N)(0.561m)=10.44Nm[/tex]
The moment of inertia is given by:
[tex]I=\frac{1}{2}M_pr^2[/tex] (3)
Mp: mass of the pulley = 8kg
[tex]I=\frac{1}{2}(8kg)(0.561m)^2=1.25kg.m^2[/tex]
You solve the equation (1) for α and replace the values of the moment of inertia and the torque to obtain the angular acceleration:
[tex]\alpha=\frac{\tau}{I}=\frac{10.44Nm}{1.25kgm^2}=8.35\frac{rad}{s^2}[/tex]
Next, you use the following formula to find the angular displacement:
[tex]\theta=\frac{1}{2}\alpha t^2[/tex]
[tex]\theta=\frac{1}{2}(8.35rad/s^2)(2.6s)^2=28.24rad[/tex]
Finally, you calculate the arc length traveled by the pulley, this arc length is equal to the vertical distance traveled by the bucket:
[tex]s=r\theta =(0.561m)(28.24rad)=15.84m[/tex]
The distance traveled by the bucket is 15.84m
