Please solve the following rotation problem! Show full work. Thanks! A uniform d

Question

Please solve the following rotation problem! Show full work. Thanks!

A uniform disk with mass m = 14.6 kg and radius R = 1.23 m lies in the x-y plane and is free to spin about its center at the origin. Three forces act in the +y-direction on the disk: > a force 300.0 N at the edge of the disk on the +x-axis, > a force 300.0 N at the edge of the disk on the -y-axis, and > a force 300.0 N acts at the edge of the disk at an angle theta = 37.0 degree above the x-axis. 1) What s the magnitude of the torque on the disk about the z axis due to F3? 2) What is the x-component of the net torque about the z axis on the disk? 3) What is the z-component of the net torque about the z axis on the disk? 4) If the disk starts from rest, what is the rotational energy of the disk F1 F2 after the forces have been applied for t = 2.5 s?

Explanation / Answer

Given that

A uniform disk with a mass (m) =14.6kg

The radius of the disk (R) =1.23m

1)

The magnitude of torque about the z-axis is given by T =300*1.23*cos(37) =294.696N

2)

The x-component of torque about the z-axis is T =0 (zero)

3)

The magnitude of torque about the z-axis due to the force F1 is T =300*1.23 =369N.m

Now the z component of net torque about the z axis is T =369N.m-294.696N=74.304N.m

4)

We know that torque is given by T =I*alpha

The moment of inertia is given by (I) = (1/2)mr2 =0.5*14.6*(1.23)2 =11.044kg.m2

alpha =T/I =74.304N.m/11.044kg.m2 =6.727rad/s2

we also know that the final angualr velocity is given by wf =wo+alpha*t

wf =alpha*t =(6.727rad/s2)(2.5)=16.819rad/sec

Threfore the rotational energy of the disk is given by

E =(1/2)Iw2=0.5*(11.044kg.m2)(16.819rad/sec)2=1562.192J

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