Solution: Rolling without Slipping
Here a solution to the ‘rolling without slipping‘ puzzle posted on this blog.
The motion of a rigid body is determined by its center of mass motion and its rotation about that center of mass (CoM). The motion of the center of mass is given by:
F = dP/dt = M ΔV / Δt
where M is the body’s total mass, ΔV the change in the CoM acceleration and F the total external force - in our case, the force applied by the stick during the short collision time, Δt. For the torque, we need only to take into account the perpendicular component of the force, FA:
Force components
Using a bit of trigonometry, one can deduce the torque:
IΔω=F*(h-R)*Δt
where I is the moment of inertia of the ball, I=(2/5)MR². Here’s the reasoning:
Computing the torque: a bit of trigonometry
The parallel component of the force, which is responsible for applying the torque, is Fsin(θ)R = F*(h-R).
Combining both equations and solving for the CoM velocity after impact, we get:
ΔV = 2R²Δω/(5(h-R))
Rolling without slipping will only occur when the velocity of a point on the rim of the ball is entirely due to its rolling motion; that is, when
ΔV=RΔω
Substituting this back into our expression for ΔV, we can solve for h:
h=(7/5)R
which is the desired solution.
Very good calculation for snooker.
But i’d like to try headshots towards ball.
By the way, you mean FB saying “For the torque, we need only to take into account the perpendicular component of the force”
Bcs FB is the responsible component for spinner torque in your figure.
Have good day
It seems to me, that in addressing only the torque provided by the stick, you are neglecting additional torque that results from the component of the force in the radial direction and the force from static friction.
The radial force acts on the CoM and has a component that is tangential to the bottom surface of the table providing an additional torque due to friction with the table.
Hi Ragidandy,
Yes, if I were to take into account the friction between the ball and the table the calculation would’ve been different. However, since nothing was stated on the matter, I’ll reserve the right to argue that the table is frictionless ;). Maybe next problem will have some friction involved!
@Assaf
You cannot have a ball rolling on a frictionless surface. On a frictionless surface the ball would be slipping smoothly and not rolling at all. For rolling without slipping there must be static friction.