# Basic Kinematics of Spheres

1. Dec 9, 2011

### dougla

1. The problem statement, all variables and given/known data

My physics classes were a long time ago, and I can't for the life of me remember how to solve this problem: If I have two spheres s1 and s2 touching, of mass m1 and m2, with radii r1 and r2, and centers c1 and c2, and I have a force Fout normal to the surface of s1, how do I calculate the kinematics of the two spheres? Picture here:

So there is Fout, acting on s1, and there must be F12, normal to the tangent where s1 and s2 meet acting on s2, and s21, opposite to F21, acting on s1, with F12 + F21 = 0; Beyond this, however, I am totally lost as to how to find what F12 and F21 are.

Also, there is no gravity and no friction.

2. Dec 9, 2011

### Simon Bridge

Um - is Fout normal everywhere on the surface of S1 or is it an applied force acting through the center of mass of the sphere?

If the latter, F12 is the component of Fout in the direction of c2. There will also be a perpendicular component which will tend to make the sphere's rotate.

3. Dec 9, 2011

### ehild

Write out Newton's second law for both spheres in terms of Fout and the unknown normal force F12. Write the equation of the constrain, too, that the spheres have to touch each other. Without friction, you do not need to take rotation of the spheres into account.

ehild

4. Dec 9, 2011

### dougla

Yes, Fout is an applied force acting on the center of mass of the sphere.

I don't think F12 is merely the component of Fout in the direction of c2, though obviously it's a scalar multiple of c2. Consider if Fout were coming from directly opposite S2. In that case, it would be like pushing two boxes, and thus F12 = m2 * a, with a = Fout / (m1 + m2), so F12 = Fout * m2 / (m1 + m2).

Also, since there is no friction, why would there be rotation on these spheres?

5. Dec 9, 2011

### ehild

The normal force is not known and it need not be constant during the motion. If you take the system of spheres, the motion of CM is determined by the external force: the normal force cancels.
Write all equations both for x and y directions using the angle between the external force and the radius to the common point to the spheres, and then cancel the normal force.

ehild

6. Dec 9, 2011

### dougla

Can I treat the instantaneous force as the force on a sloped block?

The constraint on this one is
$$a(x_2 - x_1) + b(y_2 - y_1) = c$$

which differentiates into

$$a(\ddot{x}_2 - \ddot{x}_1) + b(\ddot{y}_2 - \ddot{y}_1) = 0$$

which is nice and linear. Whereas the circle constraint is

$$(x_2 - x_1)^2 + (y_2 - y_1)^2 = D$$

and thus,

$$2(x_2 - x_1)(\ddot{x}_2 - \ddot{x}_1) + (\dot{x}_2 - \dot{x}_1)^2 + 2(y_2 - y_1)(\ddot{y}_2 - \ddot{y}_1) + (\dot{y}_2 - \dot{y}_1)^2 = 0$$

which is horrifying.

edit: although since they are at rest at the start, this should transform back into the original constraint, so, I guess I can.

Last edited: Dec 9, 2011
7. Dec 9, 2011

### ehild

This problem looks horrifying anyway. What was the original question?

ehild

8. Dec 9, 2011

### dougla

Actually it turns out the sphere constraint is identical to the sloped block constraint, though obviously the slope varies.

Since we know that $$(\dot{x}_2 - \dot{x}_1)^2 + (\dot{y}_2 - \dot{y}_1)^2 = 0$$

then

$$2(x_2 - x_1)(\ddot{x}_2 - \ddot{x}_1) + (\dot{x}_2 - \dot{x}_1)^2 + 2(y_2 - y_1)(\ddot{y}_2 - \ddot{y}_1) + (\dot{y}_2 - \dot{y}_1)^2 =$$
$$2(x_2 - x_1)(\ddot{x}_2 - \ddot{x}_1) + 2(y_2 - y_1)(\ddot{y}_2 - \ddot{y}_1) = 0$$

where the slopes are $$a = (x_2 - x_1), b = (y_2 - y_1)$$

This isn't a question for a class, I'm just curious about how to solve this problem.

9. Dec 9, 2011

### ehild

You have got equations following from the constraint. You also have Newton's equations between accelerations and forces. Those four equations contain the unknown normal force.

It is possible to use the coordinates of the CM and the difference between the coordinates as new variables.
The motion of the CM is easy to solve, it depends solely on the external force.
The difference between the coordinates can be expressed with an angle and with the constant distance (r1+r2).
At a certain angle, the spheres separate and the constraint does not act any more. From that instant, you have a free sphere and the other one moving under the effect of the external force.

ehild