Classical Mechanics — Part 2 MCQs with Explanations

non-central and non-conservative is the correct answer to “Consider two particles with position vectors $$\overrightarrow {{{\bf{r}}_1}} $$ and $$\overrightarrow {{{\bf{r}}_2}} $$ . The force exerted by particle 2 on particle 1 is $$\overrightarrow {\bf{F}} \left( {\overrightarrow {{{\bf{r}}_1}} ,\,\overrightarrow {{{\bf{r}}_2}} } \right) = \left( {{{{\bf{\dot r}}}_2} - {{{\bf{\dot r}}}_1}} \right)\left( {{r_2} - {r_1}} \right).$$ The force is”. This question is standard in competitive examination syllabi for Classical Mechanics.

remain at rest is the correct answer to “A particle constrained to move along the X-axis in a potential V = kx 2 , is subjected to an external time dependent force $$\overrightarrow {\bf{F}} \left( t \right).$$ Here, k is a constant, x, the distance from the origin and t is the time. At some time T, when the particle has zero velocity at x = 0, the external force is removed. The particle will”. This question is standard in competitive examination syllabi for Classical Mechanics.

2m is the correct answer to “A system of four particles is in X-Y plane of these two particles of masses m are located at (1, 1) and (-1, 1). The remaining two particles each of mass 2m are located at (1, 1) and (-1, -1). The xy component of moment of inertia tensor of the system of particles is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{1}{2}m{\dot x^2}\left( {1 + \frac{{{c^2}}}{{{x^4}}}} \right) - \frac{{mgc}}{x}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{\sqrt 3 }}{2}c$$ is the correct answer to “An electron gains energy so that its mass becomes 2m 0 . Its speed is”. This question is standard in competitive examination syllabi for Classical Mechanics.

α = $$\frac{1}{2}$$, β = 2 is the correct answer to “Let (p, q) and (P, Q) be two pairs of canonical variables. The transformation Q = q α cos βp, P = q α sin βp is canonical for”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\sqrt {\frac{{3k}}{{2m}}} $$ is the correct answer to “Two bodies of masses m and 2m are connected by spring constant, the frequency of normal mode is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$${I_1} = {I_2},\,{I_1} \ne {I_3}$$ is the correct answer to “A heavy symmetrical top is rotating about its own axis of symmetry (Z-axis). If $${I_1},\,{I_2}$$ and $${I_3}$$ are the principal moments of inertia along X, Y and Z axes respectively then”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$2\pi \sqrt {\frac{{2R}}{g}} $$ is the correct answer to “A hoop is pivoted at a point on circumference. The period of small oscillations in the plane of loop is”. This question is standard in competitive examination syllabi for Classical Mechanics.

harmonic oscillator is the correct answer to “The Lagrangian of a system is given by $$L = \frac{1}{2}{\dot q^2} + q\dot q - \frac{1}{2}{q^2}$$ It describes motion of a”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{v}{{\sqrt 2 }}$$ is the correct answer to “A particle of mass M moving in a straight line with speed v collides with a stationary particle of the same mass. In the centre of mass coordinate system, the first particle is deflected by 90°. The speed of second particle after collision in laboratory system will be”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{1}{7}a$$ is the correct answer to “A car is moving with constant linear acceleration a along horizontal X-axis. A solid sphere of mass M and radius R is found rolling without slipping on the horizontal floor of the car in the same direction as seen from an inertial frame outside the car. The acceleration of the sphere in the inertia frame is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{m}{2}{\left( {\frac{{dq}}{{dt}} - q} \right)^2}$$ is the correct answer to “The Hamiltonian of a particle is $$H = \frac{{{p^2}}}{{2m}} + pq,$$ where q is generalised coordinate and p is the corresponding canonical momentum. The Lagrangian is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\theta = {\tan ^{ - 1}}\left( {\gamma \cot {\theta _0}} \right)$$ is the correct answer to “A rod of length L 0 makes an angle θ 0 with the Y-axis in its rest frame while the rest frame moves to the right along the X-axis with relativistic speed v with respect to lab frame. If $$\gamma = {\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right)^{ - \frac{1}{2}}},$$ the angle in the lab frame is”. This question is standard in competitive examination syllabi for Classical Mechanics.

p = (1 - μ)mv is the correct answer to “A particle of mass m moves in a potential V(x) = $$\frac{1}{2}$$ mω 2 x 2 + $$\frac{1}{2}$$ mμv 2 , where x is the position coordinate, v is the speed and ω, μ are constants. The canonical momentum of the particle is”. This question is standard in competitive examination syllabi for Classical Mechanics.

the motion is confined in a plane is the correct answer to “For a particle moving in a central field,”. This question is standard in competitive examination syllabi for Classical Mechanics.

-80 $${\bf{\hat k}}$$ is the correct answer to “A particle of mass 2 kg is moving such that at time t second. Its position in metre is given by $$\overrightarrow {\bf{r}} \left( t \right) = 5{\bf{\hat i}} - 2{t^2}{\bf{\hat j}}.$$ The angular momentum of the particle at t = 2 s about the origin in kg-m 2 /s, is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$m\dot r{\text{ and }}m{r^2}\dot \theta $$ is the correct answer to “The Lagrangian of a particle moving in a plane under the influence of central potential is given by $$L = \frac{1}{2}m\left( {{{\dot r}^2} = {r^2}{{\dot \theta }^2}} \right) - V\left( r \right).$$ The generalised momenta corresponding to r and θ are given by”. This question is standard in competitive examination syllabi for Classical Mechanics.

px = mvx - ay and py = mvy + ax is the correct answer to “The Lagrangian of a particle of mass m moving in a plane is given by L = $$\frac{1}{2}$$ [m(v x 2 + v y 2 )] + a(xv y - yv x ) where, v x and v y are velocity components and a is constant, The canonical momenta of the particle are given by”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$L = \frac{m}{2}\left[ {{{\left( {\frac{{d{x_1}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_2}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_3}}}{{dt}}} \right)}^2}} \right] - \frac{k}{2}{\left( {{x_1} - {x_2} - l} \right)^2} + \frac{k}{2}\left( {{x_3} - {x_2} - l} \right)$$ is the correct answer to “Three particles of mass m each situated at x 1 (t), x 2 (t) and x 3 (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”. This question is standard in competitive examination syllabi for Classical Mechanics.

energy and z-component of linear momentum is the correct answer to “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”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$ \pm {\left( {\frac{k}{{2m}}} \right)^{\frac{1}{2}}}$$ is the correct answer to “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 x 1 , x 2 are displacements from equilibrium position and k > 0. The normal frequencies are”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{1}{{2m}}{\left( {\overrightarrow {\bf{p}} - \frac{q}{c}\overrightarrow {\bf{A}} } \right)^2} + q\phi $$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Classical Mechanics.

The angular momentum of the mass is zero about the equilibrium point and its Lagrangian $$\frac{1}{2}m{{\dot x}^2} - \frac{1}{2}\left( {{k_1} + {k_2}} \right){x^2}$$ is the correct answer to “A mass m is connected on either side with a spring each of spring constants k 1 and k 2 . 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”. This question is standard in competitive examination syllabi for Classical Mechanics.

angular momentum is conserved is the correct answer to “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”. This question is standard in competitive examination syllabi for Classical Mechanics.