Conservation of Momentum and Energy for a System of Connected Particles

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In summary, a system of three particles, with Q having a mass of 2m and P and R having a mass of m each, is connected by light inextensible strings of length a and placed on a smooth horizontal table. Q is projected in the positive x-direction with speed u, causing the system to move. The conservation of linear momentum and energy can be expressed in terms of the coordinates x (displacement of Q) and theta (angle of the strings). It can be shown that theta satisfies the equation (theta-dot)^2=(u^2/a^2)*(1/(2-cos^2(theta)). The equations for conservation of energy and momentum are used to derive this result.
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Homework Statement



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))


Homework Equations



equations for conservation of energy
equation for conservation of momemtum.


The Attempt at a Solution



F [tex]\cdot[/tex] x-hat=0
p [tex]\cdot[/tex]x-hat=0

p[tex]\cdot[/tex]x-hat=(m1*v1+m2*v2) [tex]\cdot[/tex] 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 ?
 
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anyone have a hard time comprehending my solution? Just say so.
 

What is conservation of momentum and energy for a system of connected particles?

Conservation of momentum and energy for a system of connected particles is a fundamental principle in physics that states that the total momentum and energy of a closed system remains constant over time, even as individual particles within the system may interact and exchange momentum and energy with each other.

How is momentum conserved in a system of connected particles?

Momentum is conserved in a system of connected particles through the principle of action and reaction. When two particles within the system interact, they exert equal and opposite forces on each other, resulting in a transfer of momentum. This transfer keeps the total momentum of the system constant.

What about energy conservation in a system of connected particles?

Energy conservation for a system of connected particles works in a similar way to momentum conservation. As the particles interact and exchange energy with each other, the total energy of the system remains constant. This is because energy cannot be created or destroyed, only transferred from one form to another.

What are some real-life examples of conservation of momentum and energy in a system of connected particles?

One common example is a collision between two objects, such as a pool ball hitting another pool ball. The total momentum and energy of the system (the two pool balls) remains constant, even though individual particles (the pool balls) may experience changes in momentum and energy during the collision.

How does the conservation of momentum and energy for a system of connected particles relate to Newton's Laws of Motion?

The conservation of momentum and energy for a system of connected particles is closely related to Newton's Laws of Motion. In particular, it relates to Newton's Third Law, which states that for every action, there is an equal and opposite reaction. This principle is essential for explaining how momentum and energy are conserved in a system of connected particles.

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