# Minimum velocity for particle to lose contact

• Tanya Sharma
In summary, the homework statement asks for the limit velocity of an object as it moves over another object. If the object is in contact with the ground, the CM moves with a speed of 2 m/s towards the left.
Tanya Sharma

## The Attempt at a Solution

Honestly I have very little idea about this problem . The contact force has to vanish for particle B to loose contact .Not sure how to relate the velocity of upper particle with the normal contact force of the lower particle .

I don't think any conservation laws would work.

I would be grateful if somebody could help me with the problem.

#### Attachments

• rod.GIF
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Assuming that the rope have constant length and the B particle can move only horizontally, this problem must ask for the limit velocity of A just it is over B particle. If A move to right, the distance go to be larger that impossible, so B start to move with some acceleration. Think about what can be this acceleration.

Or, change frame, take origin to A particle so the problem is the same that to give horizontal velocity ##-v## to B particle. The rope is constant (again).

If B loses contact, there is no force between B and the ground any more. What happens to the center of mass? Where is B relative to the center of mass, where is it relative to the ground?

It is a limit process ##t\to0##.

mfb said:
If B loses contact, there is no force between B and the ground any more. What happens to the center of mass?

The CM moves with speed v/2 towards right . The only external force acting on the system would be gravitational force .

mfb said:
Where is B relative to the center of mass,

B moves with speed v/2 towards left relative to CM .

mfb said:
where is it relative to the ground?

Since it just looses contact , it is effectively at rest relative to the ground .

I haven't solved the problem, but would start as follows.

Remember, motion of a rigid body can be considered as motion of the CM with acceleration determined by the external forces and rotation about the CM according to the external torques.
At t=0, the system has some momentum and angular momentum.
You can write the motion of ball B in terms of the motion of the CM and the rotation about it. B should stay on the ground or lift from it: its vertical acceleration can not be negative even without a normal force.

Sorry , I can't see how to implement the above strategy to deal with this problem.

If B were to stay in contact with the floor, what would be the height, ##y##, of the CM of the system as a function of angle of rotation of the system?

Can you use this relation to find an expression for ##\ddot{y}##?

Tanya Sharma
Taking upward positive and ##\theta## to be angle made by rod with the vertical .

##N - 2mg =2m\ddot{y}##

##\ddot{y} = -\frac{l}{2}[cos\theta \dot{\theta}^2+sin\theta\ddot{\theta}]##

What doe the second equation reduce at the time of interest?

Tanya Sharma
Are you getting v = 2m/s ?

My answer is in terms of the length of the rod and g. I haven't yet put in numbers. Let's see...

Yes, I get 2 m/s.

Tanya Sharma
##v=\sqrt{2gl}## . Right ?

Yep.

Tanya Sharma
I have been thinking about this problem for last two days and here you come and make this problem look so ordinary .

How do you think so clearly ?

I don't know. It always starts out kind of foggy and then the fog lifts (sometimes ).

You got the answer very quickly after I gave a hint. So, I think I might have given too much away. It's always difficult to find the right nudge.

Tanya Sharma
Thank you very very much .

Good night .

## 1. What is the "minimum velocity for particle to lose contact"?

The minimum velocity for particle to lose contact is the minimum speed at which a particle must be moving in order to break free from a surface or another particle.

## 2. How is the "minimum velocity for particle to lose contact" calculated?

The minimum velocity for particle to lose contact is calculated using the formula v = √(μgR), where v is the minimum velocity, μ is the coefficient of friction, g is the acceleration due to gravity, and R is the radius of the particle.

## 3. What factors affect the "minimum velocity for particle to lose contact"?

The factors that affect the minimum velocity for particle to lose contact include the coefficient of friction, the surface material, the shape and size of the particle, and the presence of any external forces.

## 4. Why is knowing the "minimum velocity for particle to lose contact" important in science?

The minimum velocity for particle to lose contact is important in science because it helps us understand and predict the behavior of particles in different systems, such as in fluid dynamics, material science, and particle physics.

## 5. How can the "minimum velocity for particle to lose contact" be applied in real-world scenarios?

The understanding of the minimum velocity for particle to lose contact can be applied in real-world scenarios, such as designing machinery, predicting the movement of particles in a fluid, or determining the stability of a structure or material.

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