Conceptual Questions on Mechanics

[Cube Equation]

Hi everyone

Thanks for viewing this thread. Hope it's in the right forum. I have some questions on basic mechanics that could really do with some clarifications, so I'd really appreciate any help you could provide.

1. Consider a ball with sufficient initial speed such that it moves in a vertical circular track. At 90 degrees from the base of the track, its downward gravitational force is directed tangentially along the track, which is vertical at that point. But there seems to be no interaction between the ball and the track wall. So what is the normal force and thus the centripetal force on the ball at this point?

2. Does the elasticity of a collision depend solely on the mechanical properties of the colliding object?

3. On a related note, when a ball collides with and deforms a surface of putty, is there potential energy stored in the deformed state? If so, when one reverses the deformation by performing work on the putty surface (which would be positive since the putty moves in the direction of the applied force), does this energy dissipate into heat?

And what happens when this ball collides with and deforms a surface of sand? Is all the kinetic energy converted into thermal energy?

4. Consider a rigid object free to rotate without any specific axis defined by the presence of axles. Forces are applied to provide a net torque. Does the object rotate about an axis through the center of mass due to it being associated the lowest moment of inertia for the object? If so, how does this change when there is an axle through an arbitrary point which may not be at its center of mass?

5.

.https://www.physicsforums.com/attachments/81845

I'm having some problem with understanding why the static friction in the second case must necessarily be directed to the right. My understanding is that the frictional force would act such that it opposes the slipping of the contact point and that if the magnitudes of the force and the torque were in the correct ratio such that the acceleration of center of mass were equal to the product of the radius of the yo-yo and its angular acceleration, a static frictional force would not even exist. What is erroneous with my interpretation>

6. Lastly, with the typical example of a figure skater demonstrating the conservation of angular momentum, how is energy conserved when the skater extends and retracts his/her arms? I could see that it would have something to do with the work performed by the action of his/her arms but I'm not sure how this would be able to decrease rotational kinetic energy when the skater extends the arms. The work still seems to be positive as it moves in the direction of the force.

Thanks for your time.

 

[Delta2]

i will try to answer some of your questions:

1) There is normal force at the point of theta=90, though i know our intuition tell us that there shouldn't be any (because the velocity is parallel to the track surface and the weight also the same). Still from Newton's 2nd law we know there should be a net force applied to the particle (because it has centripetal acceleration a=V^2/R and F=ma) and we have to accept this force is the normal force from the track, there is simply no other mysterious or spooky force that can come into play. It is just one case that our intuition fools us, that's all, our intuition isn't something that tell us always the truth and the absolute truth.

4) If the forces are such that they have zero net total force (but this does not necessarily imply zero net Torque) then we know that velocity of the c.o.m is constant (which means that the c.o.m is either stationary or moving in a straight line) and the rigid object clearly rotates about the c.o.m. Now If we put a stationary axle at an arbitrary point then essentialy this axle exerts a force to the body such as to keep the velocity at the point of body that the axle passes through, constant (that is constant and zero if the axle is kept stationary) so the body rotates around that point.

6) In this example its only conservation of angular momentum that holds, conservation of rotational kinetic energy doesn't necessarily holds. When the skater extends the arms angular momentum is conserved according to $I_1\omega_1=I_2\omega_2$ but the rotational kinetic energy is not conserved (if $I_2>I_1$ as such is the case when the arms are extended the moment of inertia is increased) as we can see if we do the math. Some of the rotational kinetic energy goes to heat. When the skater retracts the arms angular momentum is conserved the rotational kinetic energy is increased by the work of internal forces.

 

[Vatsal Sanjay]

I will answer a few
2. No elasticity of a collision does not "only" depend on the mechanical properties of the substance. There is something called the coefficient of restitution ($e$). It is 1 for perfectly elastic collision and <1 for inelastic collision. $e$ represents the ratio of relative velocities, or impulses, or energies (depending on the definition used) before and after the impact of two colliding entities. Now this $e$ depends not only on the mechanical properties but also on the kinematic factors leading into the collision.
3. Second part
No not just thermal. The configuration of sand particles (no matter how loosely packed) does change. So part of kinetic energy of the ball goes there. Then sound (vibrational kinetic energy of the air molecules nearby) is also there. And of course there is thermal energy.