Classical Mechanics MCQs Question and Answer in Engineering Physics Page-7 section-1

61.
The Lagrangian of a particle' of mass m is $$L = \frac{m}{2}\left[ {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left( {\frac{{dz}}{{dt}}} \right)}^2}} \right] - \frac{V}{2}\left( {{x^2} + {y^2}} \right) + W\sin \omega t$$
where V, W and ω are constants, then conserved quantities are

A. energy and z-component of linear momentum

B. energy and z-component of angular momentum

C. z-component of both linear and angular momentum

D. energy and z-component of both angular and linear momentum

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62.
The Lagrangian of a diatomic molecule is given by $$L = \frac{m}{2}\left( {\dot x_1^2 + \dot x_2^2} \right) - \frac{k}{2}{x_1}{x_2}$$ where, m is the mass of each atom and x₁, x₂ are displacements from equilibrium position and k > 0. The normal frequencies are

A. $$ \pm {\left( {\frac{k}{m}} \right)^{\frac{1}{2}}}$$

B. $$ \pm {\left( {\frac{k}{m}} \right)^{\frac{1}{4}}}$$

C. $$ \pm {\left( {\frac{k}{{2m}}} \right)^{\frac{1}{4}}}$$

D. $$ \pm {\left( {\frac{k}{{2m}}} \right)^{\frac{1}{2}}}$$

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63.
A planet moves around the sun in an elliptical orbit with semi-major axis a and time period T. T is proportional to

A. a²

B. $${{\text{a}}^{\frac{1}{2}}}$$

C. a³

D. $${{\text{a}}^{\frac{3}{2}}}$$

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64.
A particle of charge q, mass m and linear momentum $$\overrightarrow {\bf{p}} $$ enters an electromagnetic field of vector potential $$\overrightarrow {\bf{A}} $$ and scalar potential $$\phi $$. The Hamiltonian of particle is

A. $$\frac{{{p^2}}}{{2m}} + q\phi + \frac{{{A^2}}}{{2m}}$$

B. $$\frac{1}{{2m}}{\left( {\overrightarrow {\bf{p}} - \frac{q}{c}\overrightarrow {\bf{A}} } \right)^2} + q\phi $$

C. $$\frac{1}{{2m}}{\left( {\overrightarrow {\bf{p}} - \frac{q}{c}\overrightarrow {\bf{A}} } \right)^2} + \overrightarrow {\bf{p}} \cdot \overrightarrow {\bf{A}} $$

D. $$\frac{{{p^2}}}{{2m}}q\phi - \overrightarrow {\bf{p}} \cdot \overrightarrow {\bf{A}} $$

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65.
A mass m is connected on either side with a spring each of spring constants k₁ and k₂. The free ends of springs are tied to rigid supports. The displacement of the mass is x from equilibrium position.

Which one of the following is true?

A. The force acting on the mass is $${\left( {{k_1}{k_2}} \right)^{\frac{1}{2}}}x$$

B. The angular momentum of the mass is zero about the equilibrium point and its Lagrangian $$\frac{1}{2}m{{\dot x}^2} - \frac{1}{2}\left( {{k_1} + {k_2}} \right){x^2}$$

C. The total energy of the system is $$\frac{1}{2}m{{\dot x}^2}$$

D. The angular momentum of the mass is $$mx\dot x$$ and Lagrangian of system is $$\frac{m}{2}\dot x + \frac{1}{2}\left( {{k_1} + {k_2}} \right){x^2}$$

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66.
If a particle moves outwards in a plane along a curved trajectory described by r = aθ, θ = ωt, where a and ω are constants, then its

A. kinetic energy is conserved

B. angular momentum is conserved

C. total momentum is conserved

D. radial momentum is conserved

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Classical Mechanics MCQs Question and Answer in Engineering Physics

Answer & Solution

62. The Lagrangian of a diatomic molecule is given by $$L = \frac{m}{2}\left( {\dot x_1^2 + \dot x_2^2} \right) - \frac{k}{2}{x_1}{x_2}$$ where, m is the mass of each atom and x1, x2 are displacements from equilibrium position and k > 0. The normal frequencies are

Answer & Solution

63. A planet moves around the sun in an elliptical orbit with semi-major axis a and time period T. T is proportional to

Answer & Solution

64. A particle of charge q, mass m and linear momentum $$\overrightarrow {\bf{p}} $$ enters an electromagnetic field of vector potential $$\overrightarrow {\bf{A}} $$ and scalar potential $$\phi $$. The Hamiltonian of particle is

Answer & Solution

65. A mass m is connected on either side with a spring each of spring constants k1 and k2. The free ends of springs are tied to rigid supports. The displacement of the mass is x from equilibrium position. Which one of the following is true?

Answer & Solution

66. If a particle moves outwards in a plane along a curved trajectory described by r = aθ, θ = ωt, where a and ω are constants, then its

Answer & Solution

62.
The Lagrangian of a diatomic molecule is given by $$L = \frac{m}{2}\left( {\dot x_1^2 + \dot x_2^2} \right) - \frac{k}{2}{x_1}{x_2}$$ where, m is the mass of each atom and x₁, x₂ are displacements from equilibrium position and k > 0. The normal frequencies are

63.
A planet moves around the sun in an elliptical orbit with semi-major axis a and time period T. T is proportional to

64.
A particle of charge q, mass m and linear momentum $$\overrightarrow {\bf{p}} $$ enters an electromagnetic field of vector potential $$\overrightarrow {\bf{A}} $$ and scalar potential $$\phi $$. The Hamiltonian of particle is

65.
A mass m is connected on either side with a spring each of spring constants k₁ and k₂. The free ends of springs are tied to rigid supports. The displacement of the mass is x from equilibrium position.

Which one of the following is true?

66.
If a particle moves outwards in a plane along a curved trajectory described by r = aθ, θ = ωt, where a and ω are constants, then its