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In this example, we will examine a rigid body problem in which a circular disk rolls along the ground.
A circular disk with a mass of 25 kg has a concentric circular groove cut into it.
The radius of the groove, r1, is 75 millimeters and the outer radius of the disk, r2, is 200 millimeters.
The radius of gyration, k, is 175 millimeters.
The kinetic and static coefficients of friction between the disk and ground are 0.08 and 0.10, respectively.
A steady force T is applied at an angle theta relative to the horizontal to a cord that is wrapped around the circular groove.
If T is 50 Newtons and theta is 30 degrees, what are the acceleration of the disk’s center of mass, the angular acceleration of the disk,
alpha, and the friction force the ground exerts on the disk.
Start by drawing a free-body diagram of the disk.
Identify your coordinate system and label the positive direction of your coordinates.
In this case a rectangular coordinate system is more convenient.
Indicate the positive direction of rotation.
Draw the external forces acting on the disk.
The weight acts at the center of mass and points downward. Its magnitude is the mass of
the disk, m, times the gravitational acceleration, g.
The normal force the surface exerts on the disk, N, acts perpendicular to the surface.
The tension in the cable acts at the groove at an angle theta relative to the horizontal.
The friction will oppose the motion of the disk. For now we will assume friction acts in the negative x direction.
Next, draw the equivalent kinetic diagram.
Draw the components of the acceleration of the center of mass and draw the moment about the center of mass, ICM times alpha.
We are ready to apply the 3 equations of motion:
Newton’s second law in the x direction, Newton’s second law in the y direction, and the moment equation about the center of mass.
For the x direction, the x component of the tension acts in the positive x direction and friction acts in the negative x direction.
This is equation 1.
For the y direction, the y component of the tension acts in the positive y direction, the normal force acts in the positive y direction, and
the weight acts in the negative y direction.
This is equation 2.
If the y component of the tension were greater than the weight, the disk would be accelerate upward.
However, that is not the case in this problem and the disk remains on the ground.
The acceleration of the center of mass in the y direction is zero.
For the moment equation, the moment of inertia about the center of mass, ICM, is the mass times the radius of gyration squared.
Next we determine the moments of the external forces about the center of mass.
The tension produces a positive moment and has a moment arm of r1.
Friction produces a negative moment and has a moment arm of r2.
The weight and normal force produce no moment because their moment arms are 0.
We now have equation 3.
We have used the 3 equations of motion, but have 4 unknowns: f, N, alpha, and the x
component of the acceleration of the center of mass.
If rolling without slipping occurs, we can obtain a fourth equation. In that case, the x component of the center of mass is equal to negative alpha
times r2.
The negative sign appears because a positive angular acceleration would result in the center of mass accelerating in the negative x direction.
We will solve the three equations of motion assuming the disk rolls without slipping
and then calculate the friction required for this to occur.
In this case, the value of f is 29.4 Newtons.
The maximum friction force that the ground can exert on the disk is the coefficient of static friction times the normal force
Solving the three equations of motion gives a normal force of 220.3 N
So the maximum friction force is 22.03 Newtons.
This value is below the friction force required for the disk to roll without slipping, so the disk rolls and slips.
This means the correct expression for the friction force is the coefficient of kinetic friction times the normal force.
This is equation 4.
Together with the three equations of motion,
we now have 4 equations and 4 unknowns.
The acceleration of the center of mass is 1.03 meters per second squared to the right.
The angular acceleration is 0.295 radians per second squared in the counter clockwise direction.
The friction force exerted by the ground on the disk is 17.6 Newtons to the left.