Three particles of mass m each situated at x1(t), x2(t) and x3(t) respectively are connected by two spring constants k and unstretched lengths $$l$$. The system is free to oscillate only in one-dimension along the straight line joining all the three particles. The Lagrangian of the system is
A. $$L = \frac{m}{2}\left[ {{{\left( {\frac{{d{x_1}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_2}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_3}}}{{dt}}} \right)}^2}} \right] - \frac{k}{2}{\left( {{x_1} - {x_2} - l} \right)^2} + \frac{k}{2}\left( {{x_3} - {x_2} - l} \right)$$
B. $$L = \frac{m}{2}\left[ {{{\left( {\frac{{d{x_1}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_2}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_3}}}{{dt}}} \right)}^2}} \right] - \frac{k}{2}{\left( {{x_1} - {x_3} - l} \right)^2} + \frac{k}{2}\left( {{x_3} - {x_2} - l} \right)$$
C. $$L = \frac{m}{2}\left[ {{{\left( {\frac{{d{x_1}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_2}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_3}}}{{dt}}} \right)}^2}} \right] - \frac{k}{2}{\left( {{x_1} - {x_2} - l} \right)^2} - \frac{k}{2}{\left( {{x_3} - {x_2} + l} \right)^2}$$
D. $$L = \frac{m}{2}\left[ {{{\left( {\frac{{d{x_1}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_2}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_3}}}{{dt}}} \right)}^2}} \right] - \frac{k}{2}{\left( {{x_1} - {x_2} - L} \right)^2} - \frac{k}{2}{\left( {{x_3} - {x_2} - l} \right)^2}$$
Answer: Option A
This Question Belongs to Engineering Physics >> Classical Mechanics
A. increases till mass falls into hole
B. decreases till mass falls into hole
C. remains constant
D. becomes zero at radius r1, where 0 < r1 < r0
A. $$\frac{c}{3}$$
B. $$\frac{{\sqrt 2 }}{3}c$$
C. $$\frac{c}{2}$$
D. $$\frac{{\sqrt 3 }}{2}c$$
The Hamiltonian corresponding to the Lagrangian $$L = a{{\dot x}^2} + b{{\dot y}^2} - kxy$$ is
A. $$\frac{{{p_x}^2}}{{2a}} + \frac{{{p_y}^2}}{{2b}} + kxy$$
B. $$\frac{{{p_x}^2}}{{4a}} + \frac{{{p_y}^2}}{{4b}} - kxy$$
C. $$\frac{{{p_x}^2}}{{4a}} + \frac{{{p_y}^2}}{{4b}} + kxy$$
D. $$\frac{{{p_x}^2 + {p_y}^2}}{{4ab}} + kxy$$
A. circular
B. elliptical
C. parabolic
D. hyperbolic
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