Relation between moment of inertia and acceleration

[MinaHany]

Homework Statement

I am trying to find an equation relating the acceleration (not angular acceleration) to the moment of Inertia.

I have a question that says a coin and a ring have same mass and same radius, which one would reach the bottom first if they were released from the top of an incline.

I know that the higher the moment of inertia the lower the acceleration but how do I prove it?

Homework Equations

I searched for this and I have an equation:
g/1+(I/mr^2)
but I have no idea from where this equation comes.

 

[Doc Al]

Just apply Newton's laws to the rolling object. Apply it twice: Once for translation and once for rotation. Solve them together to get the acceleration.

Start by drawing a free body diagram of the forces acting on the object.

 

[MinaHany]

Thank you for the quick reply Doc Al.
I just need some help on the way..

All I know is the component of weight acting on the body down the incline is mgsin(theta)
I've never drawn a free body diagram with rotational motion before..
I know that the torque is also applied on the body, but how do I solve the two together?

F=ma and T=I(alpha)
a=r(alpha) and I=mr^2

so F+T equals what?

 

[Doc Al]

All I know is the component of weight acting on the body down the incline is mgsin(theta)

OK. The weight is one force, which you can break into components.

I've never drawn a free body diagram with rotational motion before..

Better start now. What other forces act on the body?

I know that the torque is also applied on the body, but how do I solve the two together?

What force produces the torque on the body?

 

[MinaHany]

okay so we have:
components of the weight, normal reaction from the incline, friction (because surface can't be frictionless if the body is rotating), we have the turning force producing the torque so there are acceleration and alpha down the incline. Have I missed anything?

 

[Doc Al]

okay so we have:
components of the weight, normal reaction from the incline, friction (because surface can't be frictionless if the body is rotating),

Good. Those are the forces acting on the body.

we have the turning force producing the torque

Which of the above identified forces produces the torque?

so there are acceleration and alpha down the incline.

How are those two related?

 

[MinaHany]

the friction produces the torque i guess..
because if we have no friction we don't have a turning force and thus there is no torque.

and acceleration= r times alpha

 

[Doc Al]

the friction produces the torque i guess..
because if we have no friction we don't have a turning force and thus there is no torque.

Exactly. So write your equations.

and acceleration= r times alpha

Good.