What Is the Minimum Speed Required for a Dart to Complete a Loop After Impact?

[makeAwish]

Homework Statement

A 22.00 lead sphere is hanging from a hook by a thin wire 3.90 long, and is free to swing in a complete circle. Suddenly it is struck horizontally by a 4.50 steel dart that embeds itself in the lead sphere.
What must be the minimum initial speed of the dart so that the combination makes a complete circular loop after the collision?


The attempt at a solution

I tried finding expression for tension.
at bottom, T - mg = mv^2/r
at top, T + mg = mv^2/r
Then use conservation of momentum for perfectly inelastic collision.

but i can't solve. Can someone help me pls?

Thanks!

 

[Dadface]

The minimum speed for it to just move in a complete circle is when T at the top is zero ie when mg on its own provides the centripetal force.You need to use conservation of energy and momentum.

 

[makeAwish]

The minimum speed for it to just move in a complete circle is when T at the top is zero ie when mg on its own provides the centripetal force.You need to use conservation of energy and momentum.

hmm. why energy is conserved? i thought this is perfectly inelastic collison and energy is not conserved...

 

[Dadface]

Energy is always conserved.It is an inelastic collision but not a perfect one otherwise momentum would not be conserved.With perfectly inelastic collisions the momentum before is zero.

 

[makeAwish]

Hmm.. But how u noe momentum is conserved? Cos a 4.50 steel dart that embeds itself in the lead sphere. So i thought it is perfectly inelastic.. =x

 

[Dadface]

you can get a perfectly inelastic collision if the dart and the sphere were moving towards each other with momenta that were equal in size but opposite in direction making a resultant momentum of zero. in this case the two things would stop on impact and all of the original kinetic energy would be lost(mainly as heat energy) but energy is still conserved.With your question the sphere is initially at rest and only some of the energy will be converted to heat the rest being kinetic energy.

 

[makeAwish]

Hmm. But according to my textbook and notes, for inelastic collision, energy is not conserved. When we looking at collisions, we don't look at heat energy right? I thought we only see kinetic energy? Thats why energy can be considered as not conserved..

 

[LowlyPion]

That is correct. Kinetic energy is not conserved ... but momentum is.

So what you need to determine is what minimum speed at the bottom (from conservation of momentum) will yield enough velocity to give enough Kinetic energy to raise the combined mass of dart/ball to a height of 2 radii.

 

[Doc Al]

Hmm. But according to my textbook and notes, for inelastic collision, energy is not conserved. When we looking at collisions, we don't look at heat energy right? I thought we only see kinetic energy? Thats why energy can be considered as not conserved..

You need to treat this problem as having two steps:

1) The collision itself. Since the collision is inelastic, mechanical energy is not conserved. But, as you know, momentum is.

2) After the collision. After the collision, as the sphere (plus dart) swings up, mechanical energy is conserved.

Hint: Work backwards. Find the speed of the sphere at the top first.

 

[makeAwish]

You need to treat this problem as having two steps:

1) The collision itself. Since the collision is inelastic, mechanical energy is not conserved. But, as you know, momentum is.

2) After the collision. After the collision, as the sphere (plus dart) swings up, mechanical energy is conserved.

Hint: Work backwards. Find the speed of the sphere at the top first.

at the top, T + mg = mv^2/r
where T = 0?

so v = sqrt(gr) = 6.182m/s ??

 

[LowlyPion]

at the top, T + mg = mv^2/r
where T = 0?

so v = sqrt(gr) = 6.182m/s ??

Right. Now how much kinetic energy is that at the top?

How much energy did it lose getting there? That tells you how much energy it needs at the bottom after the collision then doesn't it?

 

[makeAwish]

how do i find the initial speed of sphere? is it zero?

 

[LowlyPion]

how do i find the initial speed of sphere? is it zero?

Yes. It's hanging right? That's before the collision of course.

 

[makeAwish]

okay thanks! i got it! :):):) thanks a lot!