00:01
All right, so let's see we have an inclined plane that is inclined at an angle theta, and we have a solid disk that has a radius r that rolls without slipping.
00:10
So if we look at the forces involved, we're going to have the weight of the disk.
00:14
We'll call it m .g.
00:16
The component of the weight along the plane is m .g.
00:19
Sine theta.
00:21
The normal force is produced by m .g.
00:24
Cosine theta.
00:26
And we'll have a frictional force at the point on the disc acting kind of.
00:30
This way if you think about it because if it's rolling the friction of force opposes the motion.
00:37
So what i have is mg sine theta or the net force is you know m .a.
00:44
This is going to be m .g.
00:45
Sine theta minus the force of friction.
00:47
Now if we look at the torques involved, the torque is going to be the moment of inertia times the angular acceleration and this is going to be provided by the force of friction times the radius.
00:58
And so if we expand this a little bit our angular acceleration we can write is the linear acceleration divided by the radius and so what we can do is write the force of friction as going to be i times a over r squared right and so if we substitute that back in here what we'll have is m a is equal to m g sine theta minus i a over r squared and so we can add this to both sides.
01:31
We'll have m plus i over r squared times a equals m g sine theta.
01:40
Now for a disk, we're just going to have that the moment of inertia is one -half m r squared by cylinder like this.
01:49
So if we divide that by r -squared, we get one -half m.
01:52
So we get a is going to be two -thirds, g times the sign of theta...