Classical Mechanics MCQs Question and Answer in Engineering Physics

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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Answer & Solution

Answer: Option A

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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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Answer & Solution

Answer: Option D

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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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Answer & Solution

Answer: Option D

Solution:

According to Kepler's third law, the square of the period of revolution (T) of a planet around the sun is proportional to the cube of the semi-major axis (a) of its elliptical orbit.

Thus, T²∝a³

Taking the square root on both sides, we get:

$$T∝ {{\text{a}}^{\frac{3}{2}}}$$

Hence, the time period (T) is proportional to $${{\text{a}}^{\frac{3}{2}}}$$ which corresponds to Option D.

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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Answer & Solution

Answer: Option B

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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.
Classical Mechanics mcq question image

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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Answer & Solution

Answer: Option B

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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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Answer & Solution

Answer: Option B

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