Rotation Problems: Finding Initial Angular Velocity and Angle Turned

[nns91]

Homework Statement

1. A ring 1.5 m in diameter is pivoted at one point on its circumference so that it is free to rotate about a horizontal axis. Initially, the line joining the support and center is horizontal. What must be the initial angular velocity be if the ring is to make just a complete revolution ?

2. A wheel starts from rest with a constant angular acceleration of 2.6 rad/s^2 and rolls for 6 s. At the end of that time. (a) Through what angle has the wheel turned ?

Homework Equations

Ei=Ef
$$\tau$$=I*$$\alpha$$

The Attempt at a Solution

1. I use conservation of energy.

E_i=(1/2)*I*$$\omega$$_i²

E_f= (1/2)*I*$$\omega$$_i²

I can calculate I= 2*M*R^2.

I substitute I into the conservation of energy equation and get R*$$\omega$$_i² - R*$$\omega$$_f²= g.

How do I continue to solve for $$\omega$$_i

2. I integrate the acceleration from 0 to 6 and get the velocity right as 15.6 rad/s. Then to get the angle I integrate velocity from 0 to 6 and get 93.6 rade. However I am wrong. The result should be 46.8 rad. What did I do wrong ? How do I correct it ?

 

[nns91]

What should I substitue for omega final ??

 

[Newton V]

what other varibles you get?

 

[nns91]

That's all I get. I need to solve for omega initial though so I still don't know how to get that.

 

[nns91]

2. I tried to use the way of setting frictional force equal to mgsin$$\theta$$ and solve for$$\theta$$ but it was wrong. My teacher told me to use the non slipping condition but how do I relate that condition to angle theta ?

 

[nns91]

So I solved both of these guys successfully.

I have another problem though.

A hollow sphere and unifrom sphere of the same mass m and radius R roll down an inclined plane from the same height H without slipping. Ech is moving horizontally as it leaves the ramp. When the spheres hit the ground, the range of the hollow sphere is L. Find the range L' of the uniform sphere.

I guess I should use conservation of energy here right ?? Any hint ?? Kinda of urgent for test tomorrow.

 

[LowlyPion]

So I solved both of these guys successfully.

I have another problem though.

A hollow sphere and unifrom sphere of the same mass m and radius R roll down an inclined plane from the same height H without slipping. Ech is moving horizontally as it leaves the ramp. When the spheres hit the ground, the range of the hollow sphere is L. Find the range L' of the uniform sphere.

I guess I should use conservation of energy here right ?? Any hint ?? Kinda of urgent for test tomorrow.

You will want to consider the differences between the rotational kinetic energies - look at the moments of inertia.

 

[nns91]

Thanks. So yeah, I can find out the different in rotational kinetic. Then how should I relate that to the length each thing travel;

 

[LowlyPion]

Thanks. So yeah, I can find out the different in rotational kinetic. Then how should I relate that to the length each thing travel;

You know the relative V for each and the time to hit the ground is the same.

Since they both take the same amount of time to hit, the L/L' will be in the same ratio as V/V'

 

[nns91]

Thank you so much.