A particle of mass m is constrained to move on the plane curve xy = c (c > 0) under gravity (Y-axis vertical). The Lagrangian of the particle is given by
A. $$\frac{1}{2}m{\dot x^2}\left( {1 + \frac{{{c^2}}}{{{x^4}}}} \right) + \frac{{mgc}}{x}$$
B. $$\frac{1}{2}m{\dot x^2}\left( {1 + \frac{{{c^2}}}{{{x^4}}}} \right) - \frac{{mgc}}{x}$$
C. $$\frac{1}{2}m{{\dot x}^2}\left( {1 + \frac{c}{{{x^4}}}} \right) + \frac{{mgc}}{x}$$
D. $$\frac{1}{2}m{{\dot x}^2}\left( {1 + \frac{c}{{{x^4}}}} \right) - \frac{{mgc}}{x}$$
Answer: Option B
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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