Maximum Height of Ball on U-Shaped Ramp

[Arman777]

Homework Statement

From an İnitial Lenght h,a solid ball rolls smoothly down one side of a U-shaped ramp and then moves up the other side,which is frictionless.What maximum height does the ball reach ?

Homework Equations

Energy conservation equations

The Attempt at a Solution

İnitial Kinetic energy is mgh and final energy wlll be mgh'...Is there some lost energy ?.When the object goes down it has also rotational energy,but when it climbs it cannot rotate so there's no rotational energy so
$mgh'=\frac 1 2mv^2$ and $mgh=\frac 1 2Iω^2+\frac 1 2mv^2$ (In bottom)

from that I gain $h'=h-\frac {v^2} {5g}$ but answer says $h'=\frac {5h} {7}$

 

[TSny]

When the object goes down it has also rotational energy,but when it climbs it cannot rotate so there's no rotational energy ##

What would cause the ball to stop rotating when it starts up the other side?

 

[Arman777]

What would cause the ball to stop rotating when it starts up the other side?

no friction

 

[TSny]

Why would the absence of friction stop the rotational motion of the ball?

 

[Arman777]

there would be no torque to rotate the ball ?

 

[TSny]

If something is already rotating, you don't need a torque to keep it rotating. Think of a spinning top. If no friction acts on the top, then it will keep spinning "forever". You would need a torque to stop it rotating.

 

[TomHart]

I got 5h/7 as an answer. I couldn't follow what you were doing so I can't really help troubleshoot.

 

[Arman777]

The main idea that I stucked is where is the lost energy ? initially it is energy is $mgh$ right ?

 

[TomHart]

It looked like the two initial equations you wrote were correct. The initial potential energy is converted to linear and rotational kinetic energy. And then you showed the final potential (edit correction) energy equal to the linear kinetic energy. That is right. There is no longer any friction so the ball just keeps rotating with no friction to stop it. Therefore, that rotational kinetic energy is not lost (it just keeps going), but it does not get converted back to potential energy. So it looked to me like your thinking was correct.

 

[Arman777]

It looked like the two initial equations you wrote were correct. The initial potential energy is converted to linear and rotational kinetic energy. And then you showed the final potential (edit correction) energy equal to the linear kinetic energy. That is right. There is no longer any friction so the ball just keeps rotating with no friction to stop it. Therefore, that rotational kinetic energy is not lost (it just keeps going), but it does not get converted back to potential energy. So it looked to me like your thinking was correct.

then where am I going wrong ?

 

[haruspex]

The main idea that I stucked is where is the lost energy ? initially it is energy is $mgh$ right ?

Is the ball rotating at the start? Is it rotating at the bottom of the U? Is it rotating when it reaches its highest point on the other side?

 

[TomHart]

then where am I going wrong ?

I don't know. I can't see any in between steps.

 

[haruspex]

then where am I going wrong ?

You have left v in your answer, which is not a given variable. You need to express v in terms of h.

 

[Arman777]

I found it finally,Just one question,

here is no longer any friction so the ball just keeps rotating with no friction to stop it.

I understand this part.

Therefore, that rotational kinetic energy is not lost (it just keeps going), but it does not get converted back to potential energy.

so the change in rotational energy is zero that's why it didnt converted to potantial energy ?

 

[Arman777]

Even there's no friction ball with rotate but it will not converted to potantial energy ? .I thought it will not rotate and then derive those equations but I see that it will rotate...

 

[TomHart]

so the change in rotational energy is zero that's why it didnt converted to potantial energy ?

Right. Once the ball hits the no-friction surface, it will just keep spinning forever - well, or until it slides back down and hits the surface with friction again.

 

[Arman777]

Right. Once the ball hits the no-friction surface, it will just keep spinning forever - well, or until it slides back down and hits the surface with friction again.

I see now...withouth friction just gravaity will affect and it will came back with an acceleration (ıf its a ramp at angle $θ$ then $a=mgsinθ$)

 

[TomHart]

Yes, if there was friction on both sides of the U, the rotational energy of the ball would help to drive it upward and it would reach the original height that it started from on the original side. Without friction on the upward climb, that rotational energy never gets engaged to help with the climb.