# Trajectory of centre of mass

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1. Feb 13, 2015

1. The problem statement, all variables and given/known data
Lower surface of a plank is rough and is initially at rest on a horizontal floor. It's upper surface is smooth and has a smooth hemisphere place over it through a light string as shown. Find the trajectory of centre of mass of the sphere after the string is burnt.

2. Relevant equations
Nothing special here.
I think the question is testing some concept.
$$y_{cm}=R/2$$
X cm is zero.
{for hemisphere}

3. The attempt at a solution
Since the hemisphere is smooth,the velocity of the centre of the flat surface will be zero and its centre of mass will rotate about that point.(wrt plank).
Here since centre of mass of the system changes.so the plank will also move.
But there is friction between the plank and surface. Can you help me find its trajectory?

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2. Feb 13, 2015

### CWatters

The upper surface of the plank and the lower surface of the hemisphere are smooth - So how would the hemisphere apply a horizontal force on the plank?

3. Feb 13, 2015

The horizontal surface will exert a force because that part has friction.

4. Feb 13, 2015

### CWatters

The lower surface of the plank is rough but there is no friction between the hemisphere and the upper surface of the plank because it says...

Note that I said...

5. Feb 13, 2015

The hemisphere will just exert normal reaction force.

6. Feb 13, 2015

### CWatters

Correct. The plank and hemisphere only exert a normal/vertical force on each other.

If there is no net horizontal force acting on the hemisphere what does that mean for the motion of the hemisphere in the horizontal plane?

7. Feb 13, 2015

The hemisphere is at rest. But it's centre of mass rotates about some point.

8. Feb 13, 2015

### BvU

How did you find $y_{cm}=R/2$ ? Could you elaborate ?

9. Feb 13, 2015

I just leared the formula. the answer is straight line. but I don't know how to find thetrajectory.

10. Feb 13, 2015

I am very sorry. Y_cm = 3R/8. the other is for hemispherical shell.

11. Feb 14, 2015

### CWatters

The problem statement doesn't ask you to write any equations. You don't need to do that to solve the problem and deduce that the answer is a straight line. As you said they are just testing your understanding of a concept.

That might be correct but I suspect not in the way you are thinking.

The centre of mass is at rest before the string is cut. The net horizontal force on the hemisphere is zero. So what does that mean for the movement of the centre of mass in the horizontal plane after the string is cut? Hint: Think conservation of momentum & Newton's laws.

12. Feb 15, 2015

### CWatters

I'm away from my PC for a few days so may not get back to this thread for awhile. In the meantime....

Do you think the straight line is vertical, horizontal, or at some angle and why? If unsure see last part of my post #11.

13. Feb 15, 2015