# Integrable systems

1. Jan 28, 2009

### pentazoid

1. The problem statement, all variables and given/known data

A particle Q has mass 2m and two other particles P, R, each of mass m, are connected to Q by light inextensible strings of length a. The system is free to move on a smooth horizontal table. Initially P, Q R are at the points (0,a),(0,0),(0,-a) respectively so that they lie in a straight line with the strings taut. Q is then projected in the positive x-direction with speed u. express the conservation of linear momentum and energy for this system in terms of the coordinates x(the displacement of Q) and theta(the angle by each of the strings(.

Show that theta satisfies the equation

(theta-dot)^2=(u^2/a^2)*(1/(2-cos^2(theta))

2. Relevant equations

equations for conservation of energy
equation for conservation of momemtum.

3. The attempt at a solution

F $$\cdot$$ x-hat=0
p $$\cdot$$x-hat=0

p$$\cdot$$x-hat=(m1*v1+m2*v2) $$\cdot$$ x-hat= 2m*v_x+m(v_x+(a*theta_dot*cos(theta))= 3m(v_x)+m*a*theta_dot*cos(theta)==> 3*(v_x)+a*theta_dot*cos(theta)=0

T_1+T_2; T_1 is the kinetic energy initial and T_2 is the final kinetic energy.
V=V_1-V_2=0-m*g*a*cos(theta)
rail is smooth therefore constraint force does no work and E is convserved.

T=T_1+T_2=.5*(2m)*v_x^2+.5*m*(v_2)^2

v_2=v_2x+v_2theta

v_2= (v_x)^2+(a*theta_dot)^2+2*v_x*(a*theta_dot)*cos(theta)

T_1 would be zero since the mass is initiall at rest.

1/2*m*(3v_x^2+a^2*theta_dot^2+2*a*x_dot*theta_dot*cos(theta))-mga=mga*cos(theta)

Did I set up my conservation of energy equation correctly ?

2. Jan 29, 2009

### pentazoid

anyone have a hard time comprehending my solution? Just say so.