2 bowls rolling on an irregular surface

[mtr]

Homework Statement

2 identical bowls are rolling on 2 different routes. The first one has a convexity and the second one has a concavity in their middle parts. The convexity fits perfectly the concavity. Both bowls had the same initial velocities. Bowls are rolling without slipping and jumping. Which one will reach the goal first? Why? What can you tell about their final velocities and kinetic energy of rotational motion? Consider 2 cases:
a) there are no wastes of energy
b) there appears friction. How many times deeper or wider should be the concavity so that both bowls reach the goal simultaneously?

Homework Equations

KE = (mV^2)/2 kinetic energy
E = (Iw^2)/2 kinetic energy of rotational motion
T=fN friction

The Attempt at a Solution

The first part I did in such a way - consider an average velocities of both bowls. When the first one reaches the goal its average velocity is less than the second, because when the first one climbed the convexity its velocity decreased and thus affectioned average velocity in a negative way. The second bowl behaved just in opposite. Energies are the same, because there are no wastes.
However, I have a big problem with b). Actually, I have no idea how to solve this one.

 

[Mindscrape]

The problem is mostly about inertia. You will need to know that rotational kinetic energy is $\frac{1}{2}I\omega^2$. Think about a couple simpler cases: dropping a spherical ball down a hill, dropping a disc down a hill, and dropping a wheel down a hill, where all have the same density. Which one will finish first, or will they finish first?

 

[baba89]

I would approach this problem by first considering the basic principles of motion and energy conservation. In this scenario, we have two identical bowls rolling on different routes, one with a convexity and the other with a concavity. The bowls have the same initial velocities and are rolling without slipping or jumping.

In the first case (a), where there are no wastes of energy, both bowls will reach the goal at the same time. This is because, according to the principle of conservation of energy, the total energy of the system (bowl + surface) remains constant. The initial kinetic energy of both bowls is the same, and as they roll towards the goal, their kinetic energy of rotational motion will also be the same. This means that both bowls will have the same final velocities when they reach the goal.

In the second case (b), where there is friction, the bowls will not reach the goal at the same time. This is because friction will cause a loss of energy in the system, decreasing the kinetic energy of the bowls. The bowl on the concave surface will experience more friction than the one on the convex surface, due to the larger contact area. This means that the bowl on the concave surface will have a lower final velocity and take longer to reach the goal.

To determine how much deeper or wider the concavity needs to be for both bowls to reach the goal simultaneously, we can use the principle of energy conservation. We know that the energy lost due to friction is equal to the work done by friction, which is given by the equation T = fN, where T is the work done, f is the coefficient of friction, and N is the normal force. We can also use the equation for kinetic energy of rotational motion, E = (Iw^2)/2, where I is the moment of inertia and w is the angular velocity. By equating the energy lost due to friction to the kinetic energy of rotational motion, we can solve for the unknown variables and determine the depth or width of the concavity needed for both bowls to reach the goal simultaneously.

In summary, the bowl on the convex surface will reach the goal first in both cases, due to the nature of the surfaces and the principles of energy conservation. In the case of friction, the concave surface needs to be deeper or wider in order for both bowls to reach the goal at the same time, and this can be determined by equating the energy