How Much Velocity is Needed at Point A to Leave the Track at Point B?

[terryds]

Homework Statement

http://www.sumoware.com/images/temp/xzfrbardcbliotkt.png
[/B]
A box travels in a circular track like the picture above.
How much velocity it must have in point A to get out of the track when reaching point B ?

Homework Equations

F_centripetal = m V^2/R
E = E'

The Attempt at a Solution

First, I draw the free body diagram of the forces in point B
http://www.sumoware.com/images/temp/xzstkkpbgobpmjmi.png

Actually, I want to draw the normal force, but I don't know where the normal force direction is.
I know that the centripetal force is the force that's accelerating to the center.
So, I think that Wb sin Θ is the same as the centripetal force.
I use
Wb sin Θ= F centripetal
mg sin Θ= m v^2/r
g sin Θ= v^2/r
v = √(g r sin Θ)

Then, I use the mechanical energy conservation formula. (I assume that the point A is zero in y axis)
E = E'
1/2 mv_a^2 = mgh + 1/2 m v_b^2
1/2 v_a^2 = gh + 1/2 v_b^2
v_a^2 = 2gh + v_b^2
v_a^2 = 2g(R+R sin Θ) + v_b^2
v_a^2 = 2g(R+R sin Θ) + (√(g r sin Θ))^2
v_a^2 = 2g(R+R sin Θ) + g r sin Θ
v_a^2 = 2gR+2gRsinΘ+gr sin Θ
v_a^2 = 2gR+3gRsinΘ
v_a^2 = gR (2+3sinΘ)
v_a = √(gR (2+3sinΘ))

But, I'm not sure my answer is right.
What I doubt is when I think the W sinΘ equals centripetal force and when I don't know where the normal force direction is pointing. Actually, I want to use ∑F = ma , but I just know the weight force to draw.
Please help me.

 

[haruspex]

Actually, I want to draw the normal force, but I don't know where the normal force direction is.

At the point where the box is just falling out of the track, what do you think the magnitude of the normal force will be?

 

[terryds]

At the point where the box is just falling out of the track, what do you think the magnitude of the normal force will be?

Hmm... I think it is the same as W but with opposite direction ? Or maybe W sin theta ?

 

[haruspex]

Hmm... I think it is the same as W but with opposite direction ? Or maybe W sin theta ?

If there's still all that force exerted between the box and the track, why is it about to fall of the track?

 

[terryds]

If there's still all that force exerted between the box and the track, why is it about to fall of the track?

Maybe because the weight is bigger than the normal force, or there is no normal force..
But, i don't know how to determine the normal force in this problem

 

[haruspex]

Maybe because the weight is bigger than the normal force

Those forces both act sort-of downwards, so that isn't going to explain it.

or there is no normal force

Exactly. You lose contact with a surface when there's no force holding you to it, and so no normal force reacting.
Hence the direction of the normal force does not matter for this question, but, for future reference, if either body has a tangent plane at the point of contact then the normal force is normal to that plane. Note that if they both have tangent planes they will necessarily be the same. If neither has a tangent plane it becomes impossible to say.

 

[terryds]

Those forces both act sort-of downwards, so that isn't going to explain it.

Exactly. You lose contact with a surface when there's no force holding you to it, and so no normal force reacting.
Hence the direction of the normal force does not matter for this question, but, for future reference, if either body has a tangent plane at the point of contact then the normal force is normal to that plane. Note that if they both have tangent planes they will necessarily be the same. If neither has a tangent plane it becomes impossible to say.

Thanks for your explanation.
But, what about its centripetal force ? Does it equal W sin theta ?
And, is my answer to the problem correct ?

 

[haruspex]

Thanks for your explanation.
But, what about its centripetal force ? Does it equal W sin theta ?
And, is my answer to the problem correct ?

Yes. To keep on the track, the acceleration must be v²/r, no more, no less. The net force in the radial direction will never be less than mv²/r because the normal force will increase as necessary to prevent the block breaking into the track. But if the net force exceeds the centripetal force, even when the normal force has fallen to zero, then it will accelerate faster than v²/r radially and leave the track. The only source of such a force is g, and its radial component will be W sin(theta).