How Do You Calculate the Kinematics of Touching Spheres Under External Force?

[dougla]

Homework Statement

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 F_out normal to the surface of s₁, how do I calculate the kinematics of the two spheres? Picture here:

So there is F_out, 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.

 

[Simon Bridge]

Um - is F_out 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 F_out in the direction of c₂. There will also be a perpendicular component which will tend to make the sphere's rotate.

 

[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

 

[dougla]

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

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

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

 

[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

 

[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.

 

[ehild]

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

ehild

 

[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.

 

[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