Tomaz Kristan said:
What equations do you need? Fg=-Fr?
What forces are present here, except those two? Gravity and surface reaction?
None.
Let the overall gravitation pull on each ball be given by G_n. These can be calculated for each individual ball. Let the surface reaction on the ball on its left be given by L_n, and the surface reation on its right be given by R_n
The total force to the left on the nth ball is given by
G_n + R_n - L_n
Let the accelleration of each ball to the left be a_n. Let the mass be m_n. From F=ma we have.
m_n a_n = G_n + R_n - L_n
Let us assume for the sake of simplicity that the accelleration of every ball is equal, otherwise our surface reactions will cause problems. a_n = a
So for all n
a = \frac{G_n + R_n - L_n}{m_n}
As the accelleration is equal for all n, we have
\frac{G_n + R_n - L_n}{m_n} = \frac{G_{n+1} + R_{n+1} - L_{n+1}}{m_{n+1}}
By Newtons third law, the left surface reaction on the nth ball is equal in magnitude(but opposite in direction) to the right surface reaction on the (n+1)th ball.
L_n=R_{n+1} \text{ or } R_n = L_{n-1}
Thus,
\frac{G_n + L_{n-1} - L_n}{m_n} = \frac{G_{n+1} + L_n -L_{n+1}}{m_{n+1}}
G_n + L_{n-1} - L_n = m_n \frac{G_{n+1} + L_n -L_{n+1}}{m_{n+1}}
G_n + L_{n-1} - L_n = m_n \frac{G_{n+1} + L_n -L_{n+1}}{m_{n+1}}
L_{n-1} = L_n - G_n +m_n \frac{G_{n+1} + L_n -L_{n+1}}{m_{n+1}}
or to make it a little easier to work with, bump up the n's by one to get
L_{n} = L_{n+1} - G_{n+1} +m_{n+1} \frac{G_{n+2} + L_{n+1} -L_{n+2}}{m_{n+2}}
Starting off with the first particle (n=1), to find the total force on it we need to find the leftmost reaction force using the formula;
L_{1} = L_{2} - G_{2} +m_{2} \frac{G_{3} + L_{2} -L_{3}}{m_{3}}
We know G_2 , G_3, m_2, m_3, but we do not know L_2, L_3. Thus in order to find them we must use;
L_{2} = L_{3} - G_{3} +m_{3} \frac{G_{4} + L_{3} -L_{4}}{m_{4}}
L_{3} = L_{4} - G_{4} +m_{4} \frac{G_{5} + L_{4} -L_{5}}{m_{5}}
but we do not know L_4, L_5, so we must find those. But again their formulae will involve L_6, L_7, whose formulae involve L_8, L_9, etc, etc , etc.
In short we cannot solve the infinite system of linear equations for every L_n. Thus we cannot solve for the force, and so cannot find the accelleration of the system. Hence we find that applying mathematics to a physical problem is often more elucidating than verbal argument or consideration.
