# Rolling Objects, Friction, and Newton's Second Law

## Homework Statement

In rotational dynamics, a typical problem would be along the lines of a ring with mass m and radius r rolling down a hill with angle θ to the horizontal. Find the acceleration of the ring.

## Homework Equations

Ʃτ=I$\alpha$ = r x (friction)
ƩF = ma = mgsinθ - (friction)

## The Attempt at a Solution

These equations usually give the correct answer for the problem (plug in the numbers, isolate, solve for whatever the question asks). My question is that these equations don't seem to make sense if θ=0. That would imply that F = -(friction) and the ring is slowing down (if it was moving originally). However, that is not the case if there is no rolling friction. Why does this equation break down?

A similar dilemma I have is given a yoyo on the floor, if you pull vertically up on the string, the yoyo rolls in a certain direction. However, there is no net force horizontally, so how does it roll? If the answer is that the torque causes it, when why would newton's second law even apply in the case of yoyos rolling down a string in midair?

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## Homework Statement

In rotational dynamics, a typical problem would be along the lines of a ring with mass m and radius r rolling down a hill with angle θ to the horizontal. Find the acceleration of the ring.

## Homework Equations

Ʃτ=I$\alpha$ = r x (friction)
ƩF = ma = mgsinθ - (friction)

## The Attempt at a Solution

These equations usually give the correct answer for the problem (plug in the numbers, isolate, solve for whatever the question asks). My question is that these equations don't seem to make sense if θ=0. That would imply that F = -(friction) and the ring is slowing down (if it was moving originally). However, that is not the case if there is no rolling friction. Why does this equation break down?
It does not break down. It simply means that the static friction is zero and the ring rolls with uniform velocity. You know that static friction is not a defined force, you only know its maximum possible value.

A similar dilemma I have is given a yoyo on the floor, if you pull vertically up on the string, the yoyo rolls in a certain direction. However, there is no net force horizontally, so how does it roll? If the answer is that the torque causes it, when why would newton's second law even apply in the case of yoyos rolling down a string in midair?
The yoyo will not start to move horizontally if you pull the string exactly vertical. But it will rise a bit, detached from ground and starting to rotate...

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