Non-linear Dynamics

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In this section, we wish to cover many basic ideas of mechanics that we studied in the previous sections in much greater detail. Let's begin by analyzing what goes on when the cue is struck on certain location on the spherical surface, how much force is exerted on the cue and in what angle.

Initial Condition:
In geometry, we learned to locate a point on the spherical surface using three coordinates. For any defined force F in the angle of ascension q_F, and any defined coordinate (r, f, q) in the valid domain of values, the torque produced is:
t = Fr cosfcosq
Now using this torque equation, what is the initial angular velocity produced after the ball has been hit with force F for time t_o?
Using the impulse formula, we get w_o = at_o
at_o = t_tot(t_o/I) Where I is the moment of inertia of a solid sphere with radius r, I = (2/5)mr²
w_o = t_tot(t_o/I)


If f < p/2, t_tot = t_F + t_f
If p > p/2, then t_tot = t_F - t_f
f_s = m_smg

What is then the intial linear velocity, v_o of the ball?
Again, utilizing the impulse equation, we obtain v_cm = a_cmt_o
v_cmt_o = (Fcosq_F/m - m_kg)t_o

Slip Condition:
Once the begins to move, it may intially slip with angular velocity of only w_o.
Now we move on to generate the equations for linear velocity and angular velocity as a function of time, t.
For linear velocity, v_cm, we must integrate the sole force which causes deceleration on the ball after the initial condition is the kinetic friction, f_k = m_kmg
Hence the deceleration, a_cm = -m_kg
After integrating this expression, we obtain the general function v_cm of time t:
v_cm = v_o - m_kgt

For angular velocity, w we basically repeat the similar process, but using torque and the stationay frictional coefficient, m_s.
w = w_o + 5tt_f/(2mr²)

Non-Slip Condition:
When does the ball begin to move without slipping and what is the condition that must be met to intialize rotation without slipping?
Recall from the previous section that the corresponding linear velocity of the ball in relation to the angular velocity in non-slip case is v_cm = wr
This means that after the ball has been hit, it will begin to decelerate due to friction on the pool table and once the non-slip case of the special v_cm is met, the ball will begin to rotate without slipping. However there is an exception to this, that is when the initial torque delivered to the pool ball is at locations, q = {p/2, 3p/2} and f = p/2 which create a perfect top spin and the ball's direction will not be affected by the friction.

v_cm = wr
v_o - m_kgt = rw_o + 5tt_f/(2mr)

The time t, at which the ball begins to roll without slipping is:

How does the ball move when it is rolling without slipping? The ball will revolve on a curvature with radius of R_curve = rcot(q/2)
Bank Shots:
In real life pool, it is almost never true that angle of incidence is exactly equivalent to the angle of reflection. The angles may be similar but with friction and rotations involved, the balls sometimes bounce off the rails in completely different angles than expected.
q_o = arctan(wr/v)
As the ball hits the rail, the angular velocity produces the tangential force component of mwrt_o and normal force component of mvt_o where t_o is the length of time of impact. Hence the force due to spinning (w > 0) result in an angle q_o.