Find moment of Inertia from Force, radius and acceleration

[iva]

Homework Statement

A disc shaped object is made of a non-uniform material. Its radius is r and it is fre to rotate about an axis through its centre. If a force F applied tangentially at the edge of the object produces the angular acceleration a, what is its moment of inertia for rotation about the axis?

Homework Equations

I=mr²
F=ma

The Attempt at a Solution

If F=ma then m=F/a

so that i=mr² becomes I=Fr²/a

The answer in the book is I=Fr/a

Where did i go wrong? is it something to do with the force being tangent to the disc that i didn't do something?

Thanks!

 

[Doc Al]

If F=ma then m=F/a

Careful. "a" is the angular acceleration, not the linear acceleration. (It's better to use alpha for angular acceleration.) What's Newton's law for rotation?

so that i=mr² becomes I=Fr²/a

That formula for moment of inertia does not apply here.

All you need is Newton's 2nd law for rotation.

 

[iva]

Thanks i get it now,

So all i really needed was 1 equation:

rotational force equation: r* F=I * alpha so I=rF/alpha right?

 

[Doc Al]

Right!

 

[rwisz]

Your solution is almost correct, but there is one small mistake. The moment of inertia (I) is the measure of an object's resistance to rotational motion, and it depends on both the mass distribution and the axis of rotation. In this case, the object is rotating about an axis through its center, so we can use the equation I=mr^2. However, the mass (m) in this equation should be the total mass of the object, not just the mass at the edge where the force is applied. So the correct equation should be:

I = (m*r^2)/a

This is equivalent to the answer given in the book, I=Fr/a, since F=ma. It might seem like a small difference, but it is important to consider the total mass of the object when calculating its moment of inertia.