Consider a very gentle slope and a fairly inelastic ball. Why is it that there are some circumstances such that:
1. The ball can sit on the slope without rolling.
2. If nudged downhill, the ball will begin to roll down the slope and pick up speed.
3. If nudged uphill, the ball will roll uphill, slow down, and then start rolling downhill and pick up speed.
During (3), mustn't the ball at some point pass through whatever zero-velocity condition is required for state (1)? At that point, why doesn't the ball stop? The only explanation I can think of is that there must be some lateral motion that doesn't get zeroed.
I may be mistaken (it's been a decade since I had physics class), but I believe that physics posits that in that case, there is no actual cessation of movement (zero velocity), but rather a change in acceleration... therefore, the momentum is conserved.
The example that was given to us was a ball tossed into the air. It is tossed so that it follows exactly the same path on the way down that it traveled on the way up. Although the ball slows towards the apogee, and then accelerates away from it, in terms of mathematics there is no point at which the velocity is actually zero.
The complication comes from what you mean by the state of the ball.
Whenever a ball has zero velocity relative to a fixed (inertial) frame, you might say it has zero momentum, or zero rotational and translational energy. Its kinetic energy state is zero. But, other aspects of the ball's environment should be included in the state. Specifically, the detail that has been forgotten in this example is the external torque on the ball at the instant when it comes to rest.
A ball that is temporarily at rest on an incline experiences a downward force from gravity, essentially acting at its center. But, at that instant, the incline applies an equal and opposite upward reaction that is slightly off-center. These two forces amount to a couple (torque) that accelerates the ball back down the incline.
Thanks for covering such a wide variety of topics!
It all comes down to Newton's Laws of motion (which apply to particles with velocities much less than the speed of light with respect to an inertial frame of reference; for this problem the surface of the earth is approximately such a frame).
1. An object at rest tends to stay at rest and an object in motion tends to stay in motion unless acted upon by an external force.
2. Force is equal to the time rate of change of momentum with the derivative taken with respect to an inertial frame of reference (ie. F=ma).
3. Every action has an equal and opposite reaction.
To examine the three cases:
1. In this instance the friction force along the incline is equal to the component of the gravitational force along the incline. It is starting at rest and there are no external forces so it does not move.
2. By nudging the ball downhill the equilibrium is broken and the ball will have some momentum rolling down the hill.
3. Nudging the ball uphill imparts some velocity to the ball. The analysis can now move to conservation of energy. The ball has some initial kinetic energy based on the force of the nudge. As it travels upward that kinetic energy transfers into potential energy. At the peak, the ball will have zero velocity; all of its kinetic energy will have been transferred to potential. It will NOT have zero acceleration, and that is why it will roll down the hill. Despite the fact that at that instant there is no motion, the ball is not in an equilibrium state.
I hope that wasn't too technical an explanation. Last Friday I passed my PhD qualification exam and one of my areas was dynamics.
It's been awhile since I've taken a Physics class as well, but I think your example in #3 is not necessarily the case. On a nearly-level slope, say, 1 degree, it's entirely possible to push a ball uphill from a position of rest and have it come to a position of rest again.
The key thing is friction. For the ball to be sitting still, the frictional force acting upon the ball has to be greater than the horizontal component of the force exerted upon the ball through gravity (via sitting on a slope, rather than a flat plane).
In the abstract #3 scenario, pushing the ball reduces the frictive force, since objects that are not in motion have (if I remember correctly) stronger bonds to the surface they rest on than objects in motion, therefore the resistance they experience is greater. In the typical mental model I use to visualize this scenario, the smooth ramp surface does not exert enough frictional resistance to the ball's motion to stop the ball once the energy to move it has been imparted to it (there is a zero point in terms of kinetic/potential energy, but potential energy is still greater than resistance from friction).
However, this doesn't mean it's not possible. On a frictionless slope the ball cannot come to a resting position. On a very frictive surface it's entirely possible to push a ball uphill and have it return to a resting state.