# Rolling of an object down a slope

I have a few questions regarding the rolling of an object down a slope-

1. A ball on a slope rolls down due to a component of its weight acting on its center causing translational motion, and a frictional force at its point of contact causing a torque. How do you explain (using forces and physical principles) why a block would not roll down when it experiences the same forces as the ball?

2. I understand that static friction causes rolling without sliding, and comes into the picture to prevent sliding motion of the ball. As static friction between two surfaces takes a maximum value, is there some sort of a limit to this rolling motion beyond which the static friction becomes kinetic and the block slides along with rolling? What defines this limit (the angle of the slope's tilt or the mass of the body)?

3. In rolling without sliding, the point of contact of the ball with the surface is instantaneously at rest. Considering this point as a mass element, how can I explain it being at rest in terms of its velocity and forces? For it to be instantaneously at rest, the net force on that point would need to be 0, but how can I explain this in terms of the forces that act on that mass element specifically?

Your first question is answered here: Torque - Why does a block topple and net torque when net force is non-zero?.

1. I understand that static friction causes rolling without sliding, and comes into the picture to prevent sliding motion of the ball. As static friction between two surfaces takes a maximum value, is there some sort of a limit to this rolling motion beyond which the static friction becomes kinetic and the block slides along with rolling? What defines this limit (the angle of the slope's tilt or the mass of the body)?

Yes, there is of course a limit. For simplicity, we can consider a wheel. The limit would obviously depend on the magnitude, direction (determines whether or not the normal force between the ground and wheel increases), and point of application (determines whether a torque is applied) of the applied force, the coefficient of friction between the surfaces, the angle of the incline, and depending on your situation, other factors as well.

To simplify the calculations and give a more concrete answer, we can assume that the force we apply acts completely horizontally at the center of mass of the wheel. The reason I impose these specific constraints is because I have answered that question in detail here: The maximum force acting on a wheel that rolls without slipping as a function of the coefficient of friction.

1. In rolling without sliding, the point of contact of the ball with the surface is instantaneously at rest. Considering this point as a mass element, how can I explain it being at rest in terms of its velocity and forces?

In terms of velocity, check out my answer here: Why does a car moving in a circular track experience static friction if it is already in motion?

In terms of a force-related, you can calculate the equations of motion of a point on the rim.

To simplify the approach, assume the wheel is rolling without slipping at a constant speed (in that case the net force on the wheel is zero, but a point on the rim still experiences a force). Using just sine and cosine functions, you can find the x- and y-positions as a function of time for a point on the rim. Take the derivative of the position to get the velocity. Set velocity equal to zero. You'll realize that velocity is zero when the point on the rim touches the ground.

See the following Desmos animation: https://www.desmos.com/calculator/avz17idpcn. Notice how $$v_x$$ and $$v_y$$ are both zero whenever the orange point touches the ground.