3 balls rolling down different-shaped inclined platforms

[jaguar___]

Homework Statement

which ball reaches at the end of the board first, which ball reaches ground first, which ball would be fastest when reached ground and at the end of the board?
heights are same and balls are identical

Homework Equations

The Attempt at a Solution

i think since ball rolls there's no friction and mgh=1/2mv^2 => so the all have same speed when they reached ground and at end of board
and since in condition b ball reaches the max speed fastest it reaches first at end of the board

 

[kuruman]

Homework Statement

which ball reaches at the end of the board first, which ball reaches ground first, which ball would be fastest when reached ground and at the end of the board?
heights are same and balls are identical

Homework Equations

The Attempt at a Solution

i think since ball rolls there's no friction and mgh=1/2mv^2 => so the all have same speed when they reached ground and at end of board
and since in condition b ball reaches the max speed fastest it reaches first at end of the board

That sounds about right.

 

[haruspex]

i think since ball rolls there's no friction

No, if there were no friction the balls would slide. It is static friction, so does not reduce the total KE, but it diverts some of the KE into rotation.

all have same speed when they reached ground and at end of board

That is true because for each ball the same fraction of energy goes into rotation. The speed will be less than √(2gh), but to the same extent for each.

since in condition b ball reaches the max speed fastest

That does not follow. B starts with a gentler gradient. It could take longer to get past that than C takes for the whole descent. As against that, C has a greater horizontal distance to cover.
I cannot see that it is possible to decide merely from the diagram. The exact shape of B needs to be specified.

 

[kuruman]

That does not follow. B starts with a gentler gradient. It could take longer to get past that than C takes for the whole descent.

It could but then the question would be difficult to answer even when the details of track B are given. This is a conceptual question after all. I interpreted the intent of the question to be the realization that in B the ball loses contact with the track and is in free fall thereafter (after acquiring sufficient horizontal velocity to clear the bottom of the track). Therefore, for a good portion of the trip the vertical acceleration of B is g while for A and C the vertical acceleration is less than g during the entire trip.

 

[haruspex]

This is a conceptual question after all.

Sure, but which concept?!
It is quite possible that if the profiles were exactly as shown B would be still be approaching the drop when C reached the bottom.
I tried setting this up using a web digitising tool (https://apps.automeris.io/wpd/), but it's too sensitive to the exact slope at the start to produce a reliable result.

 

[kuruman]

Sure, but which concept?!
It is quite possible that if the profiles were exactly as shown B would be still be approaching the drop when C reached the bottom.
I tried setting this up using a web digitising tool (https://apps.automeris.io/wpd/), but it's too sensitive to the exact slope at the start to produce a reliable result.

Assuming that there is identical conversion of potential to rotational energy, I think that the main idea of the question is that mechanical energy conservation can be used to predict that the speed of an object dropping by height ##h## will be the same regardless of the path followed; additionally the question addresses the misconception that equal final speeds do not necessarily imply equal travel times.

I admire your effort to quantify the problem. My idea to this effect is to approximate the track as the superposition of two quarter-circles and a straight piece. (See yellow line below). I suspect that's how the creator of the question generated the picture. The dimensions are radius of 1/4 circle, R = 0.8 u, radius of sphere r = 0.15 u, length of straight piece h = 0.7 u. I don't have the time right now to play with it, but it's reasonable to assume that the ball starts moving from rest because it's off the vertical by angle θ = r/(R+r).

 

[haruspex]

approximate the track as the superposition of two quarter-circle

Good idea. The key then is the ratio between the radius of the curve and the radius of the ball. The greater the ratio the shallower the initial gradient and the longer the ball will take ... without limit.
I'll try using the measurements you have.

 

[kuruman]

I think I have some numerical answers. I will provide the method, but not the numbers to comply with PF rules. I assumed frictionless sliding with no rolling.
For b
1. Found the critical angle ##\theta_{crit}## at which contact is lost.
2. Found that while contact is maintained ##\frac{d\theta}{dt}=\sqrt{\frac{2g}{R+r}(\cos\theta_0-\cos\theta)}##.
3. Integrated numerically $$I_{crit}=\int_{\theta_0}^{\theta_{crit}}\frac{d\theta}{\sqrt{\cos\theta_0-\cos\theta}}$$
to obtain the value of the time required for the ball to lose contact, ##t_{crit}=\sqrt{\frac{R+r}{2g}}I_{crit}##.
4. Found the y-component of the velocity at ##t=t_c## and then used projectile kinematics to find the time of flight, ##t_f##.
5. Found the time of transit using ##T_b=t_{crit}+t_f##.

Also, just for fun I calculated that the horizontal displacement of ball b is very close to ##2R##, clearly a coincidence because the height of the vertical straight piece ##h## affects the time of flight.
Note: I assigned meters to the units previously posted so that g = 9.8 m/s² could be used without rescaling.

For c
1. Found the upper corner of the incline to be at ##(x_0,~y_0)= (2.1, 2.3)## m.
2. Calculated the down-the-incline acceleration ##a = g \sin\beta = g \frac{y_0}{\sqrt{x_0^2+y_0^2}}## and ...
3. Used it to find the time of transit ##T_c## from the kinematic equation ##\sqrt{x_0^2+y_0^2}-r=\frac{1}{2}aT_c^2.##
I found the value of ##T_b## to be about 33% higher than the value of ##T_c##. Frankly, I did not expect that.

 

[haruspex]

I assumed frictionless sliding with no rolling.

That does give C an advantage. If rolling is assumed, when the ball loses contact in B it will no longer need to acquire rotational KE, so all goes into linear KE.
On the other hand, I would say the problem setter did not intend loss of contact to be considered.