Height to which rolling ball rises on a surface

[Krushnaraj Pandya]

1. The problem
A ball moves without sliding on a horizontal surface. It ascends a curved track upto height h and returns. Value of h is h1 for sufficient rough curved track to avoid sliding and is h2 for smooth curved track, then how are h1 and h2 related (greater, lesser, equal or multiplied by some integral factor)?

2. Intuitive answer and attempt at solving mathematically
My first intuition was that friction would hinder translational motion and therefore h1<h2 but then I wondered if the ball is in puring rolling before, it would continue to do so regardless of the surface and so h1 might be equal to h2. I tried conserving energy but friction is present in one case, and even though the work done by friction will be zero I can't figure out how to find both mathematically.

P.S. the answer given is h1>h2

 

[haruspex]

intuition was that friction would hinder translational motion

Is the rate of rotation increasing or decreasing as it climbs the curve? What does that tell you about the direction of the friction?

 

[Orodruin]

I tried conserving energy but friction is present in one case, and even though the work done by friction will be zero I can't figure out how to find both mathematically.

If the work done by friction is zero, how does it affect the conservation of energy?

 

[Krushnaraj Pandya]

If the work done by friction is zero, how does it affect the conservation of energy?

I wanted to apply the work energy theorem, but that got me confused too

 

[Krushnaraj Pandya]

Is the rate of rotation increasing or decreasing as it climbs the curve? What does that tell you about the direction of the friction?

it is decreasing, to maintain pure rolling as v decreases...therefore friction must be in the forward direction- am I right?

 

[Krushnaraj Pandya]

it is decreasing, to maintain pure rolling as v decreases...therefore friction must be in the forward direction- am I right?

that'd mean friction helps the translatory motion and so h1>h2.
How do I find the heights mathematically though, I wrote 1/2 Iw^2 + 1/2 mv^2=mgh2 (conserving energy for h2) but I don't know how to find h1

 

[Krushnaraj Pandya]

that'd mean friction helps the translatory motion and so h1>h2.
How do I find the heights mathematically though, I wrote 1/2 Iw^2 + 1/2 mv^2=mgh2 (conserving energy for h2) but I don't know how to find h1

If I apply work-energy theorem I get h1=h2, I wrote change in KE as 1/2 Iw^2 + 1/2 mv^2 =mgh1, since only g is doing work on it and KE at top is 0. This is why I'm confused as to where I'm going wrong

 

[haruspex]

1/2 Iw^2 + 1/2 mv^2=mgh2 (conserving energy for h2)

What is the final rotation rate in the h2 case?

1/2 Iw^2 + 1/2 mv^2 =mgh1

Right.

 

[Krushnaraj Pandya]

What is the final rotation rate in the h2 case?

Right.

in the h2 case, there is no torque for rolling rate to change, but g is decreasing v- the ball will stop pure rolling then and start sliding?

 

[haruspex]

in the h2 case, there is no torque for rolling rate to change

So what rotational energy remains?

 

[Krushnaraj Pandya]

So what rotational energy remains?

ohh...so initial and final rotational energies are the same, only linear kinetic energy will convert to h- therefore 1/2 mv^2=mgh2. (The difference between visualizing rolling and spinning just became clear to me, Thanks)
is that correct?

 

[haruspex]

ohh...so initial and final rotational energies are the same, only linear kinetic energy will convert to h- therefore 1/2 mv^2=mgh2. (The difference between visualizing rolling and spinning just became clear to me, Thanks)
is that correct?

Right.

 

[Krushnaraj Pandya]

Right.

Thanks a lot :D