Trajectory of hemisphere's center of mass on smooth plank with friction

[AdityaDev]

Homework Statement

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.

Homework Equations

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

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?

 

[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?

 

[AdityaDev]

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?

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

 

[CWatters]

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

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...

It's upper surface is smooth and has a smooth hemisphere place over it

Note that I said...

How would the hemisphere apply a horizontal force on the plank?

 

[AdityaDev]

The hemisphere will just exert normal reaction force.

 

[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?

 

[AdityaDev]

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?

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

 

[BvU]

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

 

[AdityaDev]

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

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

 

[AdityaDev]

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

 

[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.

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

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.

 

[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.

 

[AdityaDev]

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.

straight line. the centre of mass will move vertically downwards since there is no horizontal force.

 

[CWatters]

straight line. the centre of mass will move vertically downwards since there is no horizontal force.

Correct.