Finding the Optimal Initial Speed for Circular Motion - What's the Solution?

[roman15]

this is the question.
A ball is placed against the inner edge of a cylinder at a height h=4R where R is the radius of the cylinder. What initial horizontal speed v0 tangential to the cylinder wall should be given to the ball so that it will have completed exactly 2 revolutions around the inside of the cylinder when it hits the ground?

I was completely lost when I saw this one!

 

[hikaru1221]

Is there any other information given? At least, the masses of the ball and cylinder should be provided.

 

[Mindscrape]

Yeah, it's tough to know without an explicit description of where the ball is initially placed and where the cylinder initially is. I think you may have to assume that the cylinder starts 4R off the ground, and that the ball is at the bottom of the cylinder. So then you have to figure out the angular frequency that will let it do 2 revolutions by the time the cylinder falls the 4R distance.

 

[roman15]

the only other information is to neglect friction
the ball starts at the top of the cylinder and moves down along the inner edge of the cyclinder

 

[hikaru1221]

Ah, now I can understand what the question means exactly.
Since there is no friction, the cylinder exerts only a normal force to the ball, and the normal force acts towards the center. The other force exerted on the ball is gravity, which points vertically downwards. Therefore, the horizontal component of v remains unchanged! Plus that the vertical component of v is due to gravity only, you can calculate the time period for one cycle & the time period for the ball to land on the ground, can't you?

 

[Mindscrape]

Oh, I see now too. At this point the problem is just like any other loop problem. You need to look for a period of 2.5.

 

[roman15]

ok you i kinda came to this same realization today
since there is no initial vertical component of velocity and the acceleration is only due to gravity, can't i just use projectile kinematics to solve this now
where the change in height would be 4R and the range would be 2x the circumference of the circle

 

[Mindscrape]

Why not just say that

$$v=\omega R$$ where

$$\omega = 2 \pi/ T$$

and use kinematics to solve for how long it takes the ring to fall, and thus how long your period needs to be. Do you follow?