Two Rolling Objects Racing Down a Hill?

[dwangus]

Homework Statement

There are two objects rolling down a hill of incline theta, one is a sphere and one is a disk, each of equal radius and mass.
Which one gets down first and how much faster than the other?
What's the coefficient of static friction of the hill?

Homework Equations

Moment of inertia equations for disk and sphere.
Net Torque = Iα
α = a/r
Mgsinθ
Frictional Force
Net Force = ma

The Attempt at a Solution

Ok, so I left out information on purpose, because I just need the general solutions to work these things out.
I know that the sphere will obviously win because it has a lower coefficient of moment of inertia. And I know that for each object's moment of inertia will lead to different angular accelerations, which can lead you to understanding how fast each is individually going (alpha = a/r). And I also know that the net torque force is equal to I x alpha, and that the torques needed are mgsinθ and the force of friction... but in the end, I'm not quite sure how to put the frictional force and the gravitational force together to produce angular or linear acceleration, because the gravitational force acts through each object's center of mass, and the frictional force acts on the edge? I don't know how to reconcile the different axes...

And because of that, I absolutely do not know how to find the static friction of the cliff.
Please help? You can make up your own radii and masses and angles if you want, this is all purely theoretical anyways.

 

[haruspex]

Since it is a question about rolling, you don't really care about the frictional force beyond that it is enough to prevent sliding. So the easiest way is to take moments about a point for which the frictional force has no moment. Where would that be?

 

[dwangus]

The contact point of the object and the cliff?

 

[haruspex]

The contact point of the object and the cliff?

Yes.

 

[Nickie]

I would approach this problem by first identifying all the relevant variables and equations that can be used to solve it. From the given information, we know that there are two rolling objects, a sphere and a disk, with equal radius and mass, rolling down a hill of incline theta. We also have the equations for moment of inertia, net torque, angular acceleration, net force, and frictional force.

To determine which object reaches the bottom of the hill first and how much faster, we can use the equation for net torque, which is equal to the product of moment of inertia and angular acceleration. Since both objects have equal mass and radius, their moments of inertia will also be equal. However, the sphere has a lower coefficient of moment of inertia compared to the disk, meaning it will have a higher angular acceleration. This results in the sphere reaching the bottom of the hill first and faster than the disk.

To determine the coefficient of static friction of the hill, we can use the equation for net force, which is equal to the product of mass and acceleration. The acceleration in this case is the linear acceleration of the objects as they roll down the hill. This acceleration is caused by the net force acting on them, which is the difference between the gravitational force and the frictional force. By setting the net force equal to the gravitational force, we can solve for the frictional force. The coefficient of friction can then be calculated by dividing the frictional force by the normal force, which is equal to the gravitational force in this case.

To reconcile the different axes, we can use the concept of torque. Torque is a rotational force and is defined as the product of force and the perpendicular distance from the point of rotation to the line of action of the force. In this case, the gravitational force acts through the center of mass of the objects, while the frictional force acts on the edge of the objects. By considering the moment arm, or the perpendicular distance from the point of rotation to the line of action of the force, we can calculate the torque for each force and add them together to get the net torque.

In conclusion, by using the relevant equations and considering the concept of torque, we can determine which object reaches the bottom of the hill first and how much faster, as well as calculate the coefficient of static friction of the hill.