Classical Mechanics — Part 1 MCQs with Explanations

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, T 2 ∝a 3 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 .

becomes zero at radius r1, where 0 < r1 < r0 is the correct answer to “A mass m is constrained to move on a horizontal frictionless surface. It is set in circular motion with radius r 0 and angular speed ω 0 by an applied force $$\overrightarrow {\bf{F}} $$ communicated through an inextensible thread that passesthrough a hole on the surface as shown in figure given below. Then, this force is suddenly doubled. The magnitude of the radial velocity of the mass”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{\sqrt 3 }}{2}c$$ is the correct answer to “The speed of a particle whose kinetic energy is equal to its rest mass energy, is given by (c is the speed of light in vacuum)”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{{p_x}^2}}{{4a}} + \frac{{{p_y}^2}}{{4b}} + kxy$$ is the correct answer to “The Hamiltonian corresponding to the Lagrangian $$L = a{{\dot x}^2} + b{{\dot y}^2} - kxy$$ is”. This question is standard in competitive examination syllabi for Classical Mechanics.

parabolic is the correct answer to “A particle is moving in an inverse square field. If the total energy of the particle is positive, then trajectory of particle is”. This question is standard in competitive examination syllabi for Classical Mechanics.

The maximum force of friction is μmg is the correct answer to “An object of mass m rests on a surface with coefficient of static friction μ. Which of the following is not correct”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{44}}{5}M{R^2}$$ is the correct answer to “Two solid spheres of radius R and mass M each are connected by a thin rigid rod of negligible mass. The distance between the centre is 4R. The moment of inertia about an axis passing through the centre of symmetry and perpendicular to the line joining the sphere is”. This question is standard in competitive examination syllabi for Classical Mechanics.

2Ma2ω is the correct answer to “A circular hoop of mass M and radius a rolls without slipping with constant angular speed ω along the horizontal X-axis in the X-Y plane. When the hoop is at a distance d = $$\sqrt 2 $$ a from the origin, the magnitude of the total angular momentum of the hoop about the origin is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{d{q_i}}}{{dt}} = \frac{{\partial H}}{{\partial {P_i}}}{\text{ and }}\frac{{ - d{P_i}}}{{dt}} = \frac{{\partial H}}{{\partial {q_i}}}$$ is the correct answer to “Hamiltonian canonical equations of motion for a conservation system are”. This question is standard in competitive examination syllabi for Classical Mechanics.

2 is the correct answer to “A cylinder of mass M and radius R is rolling down without slipping on an inclined plane of angle of inclination θ. The number of generalised coordinate required to describe the motion of the system is”. This question is standard in competitive examination syllabi for Classical Mechanics.

n = 1 and n = -2 is the correct answer to “A particle moves in a central force field $$\overrightarrow {\bf{F}} = k{r^n}{\bf{\hat r}},$$ where k is constant, r is distance of the particle from the origin and $${{\bf{\hat r}}}$$ is the unit vector in the direction of $$\overrightarrow {\bf{r}} $$. Closed stable orbits are possible for”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{5\pi \rho {R^5}}}{{12}}$$ is the correct answer to “The moment of inertia of a uniform sphere of radius, r about an axis passing through its centre is given by $$\frac{2}{5}\left( {\frac{{4\pi }}{3}{r^5}\rho } \right).$$ A rigid sphere of uniform mass density $$\rho $$ and radius R has two smaller spheres of radii $$\frac{R}{2}$$ hollowed out of it as shown in the figure given below. The moment of inertia of the resulting body about Y-axis is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{{m_0}c}}{{\sqrt 3 }}$$ is the correct answer to “The mass m of a moving particle is $$\frac{{2{m_0}}}{{\sqrt 3 }},$$ where m 0 is rest mass. The linear momentum of the particle is”. This question is standard in competitive examination syllabi for Classical Mechanics.

an ellipse is the correct answer to “A space station moving in a circular orbit around the earth goes into a new bound orbit by fixing its engine radially outwards. This orbit is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{91}}{{48}}m{a^2}$$ is the correct answer to “A particle of mass m is attached to a thin uniform rod of length a and mass 4m. The distance of the particle from the centre of mass of the rod is $$\frac{a}{2}.$$ The moment of inertia of the combination about an axis passing through a normal to the rod is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\left\{ {\theta ,\,\dot \theta } \right\} = 1$$ is the correct answer to “The Poisson bracket between $$\theta $$ and $${\dot \theta }$$ is”. This question is standard in competitive examination syllabi for Classical Mechanics.

Q = p and P = q is the correct answer to “If p and q are the position and momentum variables, which one of the following is not canonical transformation”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{1}{2}\left( {\overrightarrow \tau \cdot \overrightarrow \omega + \overrightarrow {\bf{L}} \cdot \overrightarrow \alpha } \right)$$ is the correct answer to “A rigid body is rotating about its centre of mass; fixed at origin with an angular velocity $$\overrightarrow \omega $$ and angular acceleration $$\overrightarrow \alpha $$. If the torque acting on it is $$\overrightarrow \tau $$ and its angular momentum is $$\overrightarrow {\bf{L}} $$, then the rate of change of its kinetic energy is”. This question is standard in competitive examination syllabi for Classical Mechanics.

energy is the correct answer to “The homogeneity of time leads to the law of conservation of (a)”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$ - 2m{u_0}\omega \hat \theta $$ is the correct answer to “A rigid frictionless rod rotates anticlockwise in a vertical plane with angular velocity $$\overrightarrow \omega $$. A bead of mass m moves outward along the rod with constant velocity $$\overrightarrow {{u_0}} $$ . The bead will experience a coriolis force”. This question is standard in competitive examination syllabi for Classical Mechanics.

$${\omega ^2} = 2{a^2}$$ is the correct answer to “The Lagrangian for a three-particle system is given by $$L = \frac{1}{2}\left( {\dot n_1^2 + \dot n_2^2 + \dot n_3^2} \right) - {a^2}\left( {n_1^2 + n_2^2 + n_3^2 - {n_1}{n_3}} \right)$$ where, a is real, then one of the normal coordinates has a frequency $$\omega $$ given by”. This question is standard in competitive examination syllabi for Classical Mechanics.

Only 1 is canonical is the correct answer to “For the given transformation 1. Q = p and P = -q 2. P = q and Q = p where p, q are canonically conjugate variables, which one of the following statements is true”. This question is standard in competitive examination syllabi for Classical Mechanics.

$${{\dot p}_\theta } = - mgL\,\sin \theta ,\,\dot \theta = \frac{{{p_\theta }}}{{m{L^2}}}$$ is the correct answer to “The Lagrangian for a simple pendulum is given by $$L = \frac{1}{2}m{L^2}{{\dot \theta }^2} - mgL\left( {1 - \cos \theta } \right)$$ Hamiltonian equations are given by”. This question is standard in competitive examination syllabi for Classical Mechanics.

Cannot be determined from the information given is the correct answer to “Two events are separated by a distance of 6 × 10 5 km and the first event occurs 1 s before the second event. The interval between two events is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\ddot x + \alpha \dot x + \frac{k}{m}x = 0$$ is the correct answer to “The lagrangian of a particle of mass m moving in one dimension is $$L = \exp \left[ {\left( {\alpha t} \right)\frac{{M{{\dot x}^2}}}{2} - k{x^2}} \right]$$ where, $$\alpha $$ and k are positive constants. The equation of motion of the particle is”. This question is standard in competitive examination syllabi for Classical Mechanics.

energy is not conserved is the correct answer to “Although mass energy equivalence of special relativity allows conversion of a photon to an electron-positron pair, such a process cannot occur in free space because”. This question is standard in competitive examination syllabi for Classical Mechanics.

the angular momentum of the particle about the centre of attraction is a constant is the correct answer to “The Lagrangian forthe Kepler problem is given by $$L = \frac{1}{2}\left[ {{{\dot r}^2} + {r^2}{{\dot \theta }^2}} \right] + \frac{\mu }{r}\,\,\,\,\,\,\,\left( {\mu > 0} \right)$$ where, $$\left( {r,\,\theta } \right)$$ denotes the polar coordinates and mass of the particle is unity, then”. This question is standard in competitive examination syllabi for Classical Mechanics.

iA (p, q) is the correct answer to “For a simple harmonic oscillator, the Lagrangian is $$L = \frac{1}{2}{{\dot q}^2} - \frac{1}{2}{q^2},\,{\text{if }}A\left( {p,\,q} \right) = \frac{{p + iq}}{2}$$ and H(p, q) is the Hamiltonian of the system, the Poisson bracket, {A(p, q), H(p, q)} is given by”. This question is standard in competitive examination syllabi for Classical Mechanics.

2mvω is the correct answer to “A bead of mass m slides along a straight frictionless rigid wire rotating in a horizontal plane with a constant angular speed ω. The axis of rotation is perpendicular to the wire and passes through one end of the wire. If r is the distance of the mass from the axis of rotation and v is its speed, then the magnitude of the Coriolis force is”. This question is standard in competitive examination syllabi for Classical Mechanics.

the total force on the system must be zero is the correct answer to “If for a system of N particles of different masses m 1 , m 2 , . . . m N with position vectors $${\overrightarrow {\bf{r}} _1},\,{\overrightarrow {\bf{r}} _2},\,.\,.\,.\,{\overrightarrow {\bf{r}} _N}$$ and corresponding velocities $${\overrightarrow {\bf{v}} _1},\,{\overrightarrow {\bf{v}} _2},\,.\,.\,.\,{\overrightarrow {\bf{v}} _N}$$ respectively such that $$\sum\limits_i {\overrightarrow {{{\bf{v}}_i}} = 0,} $$ then”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{\partial L}}{{\partial \dot \phi }}$$ is the correct answer to “The Lagrangian of a free particle in spherical polar coordinates is given by $$L = \frac{1}{2}m\left[ {{{\dot r}^2} + r{{\dot \theta }^2} + {r^2}{{\dot \phi }^2}{{\sin }^2}\theta } \right]$$ The quantity that conserved is”. This question is standard in competitive examination syllabi for Classical Mechanics.

two is the correct answer to “Two particles of equal masses are connected by an inextensible string of length L. One of the masses is constrained to moves on the surface of a horizontal table. The string passes through a small hole in the table and the other mass is hanging below the table. The only constraint is that the first mass moves on the surface of the table. The number of degree of freedom of the masses string system is”. This question is standard in competitive examination syllabi for Classical Mechanics.

6.4 × 10-6 s is the correct answer to “Assuming the mean life, of a muon (in its rest frame) to be 2 × 10 -6 s, its life time in the laboratory frame, when it is moving with a velocity 0.95 c is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$\frac{{2\left( {1 + \dot q} \right)}}{{{q^3}}}$$ is the correct answer to “A particle is moving under the action of a generalised potential $$V\left( {q,\,\dot q} \right) = \frac{{\left( {1 + \dot q} \right)}}{{{q^2}}}$$ The magnitude of generalised force is”. This question is standard in competitive examination syllabi for Classical Mechanics.

5 is the correct answer to “The moment of inertia tensor of a rigid body is given by \[I = \left[ {\begin{array}{*{20}{c}} 8&0&{ - 4} \\ 0&4&0 \\ { - 4}&0&8 \end{array}} \right]\] Moment of inertia a out an axis $${\bf{\hat n}} = \left( {\frac{1}{2},\,\frac{{\sqrt 3 }}{2},\,0} \right)$$ is”. This question is standard in competitive examination syllabi for Classical Mechanics.

ω is the correct answer to “A closed tall jar containing air and a fly placed on a sensitive weighing machine when the fly is stationary, the reading of the weighing machine is ω. If the fly starts with some upward acceleration, the reading of the machine will be”. This question is standard in competitive examination syllabi for Classical Mechanics.

$$m\sqrt {2GMb} $$ is the correct answer to “Consider a comet of mass m moving in a parabolic orbit around the sun. The closets distance between the comet and the sun is b, the mass of the sun is M and universal gravitation constant is G. The angular momentum of the comet is”. This question is standard in competitive examination syllabi for Classical Mechanics.

$${{\ddot x}_1} + {\omega _0}^2{x_1} = {\omega _0}^2\mu {x_2},\,{{\ddot x}_2} + {\omega _0}^2{x_2} = {\omega _0}^2\mu {x_1}$$ is the correct answer to “The Lagrangian of two coupled oscillators of mass m each is $$L = \frac{1}{2}\left( {{{\dot x}_1}^2 + {{\dot x}_2}^2} \right) - \frac{1}{2}m{\omega _0}^2\left( {{x_1}^2 + {x_2}^2} \right) + m{\omega _0}^2\mu {x_1}{x_2}$$ The equations of motion are”. This question is standard in competitive examination syllabi for Classical Mechanics.

The acceleration of the comet is maximum when it is closest to the sun is the correct answer to “Which one of the following is true for the above system”. This question is standard in competitive examination syllabi for Classical Mechanics.

ω, $$\sqrt 3 $$ ω and $$\sqrt 3 $$ ω is the correct answer to “A particle of mass m is moving in a potential of the form V(x, y, z) = $$\frac{1}{2}$$ mω 2 (3x 2 + 3y 2 + 2z 2 + 2xy). The oscillation frequencies of the three normal modes of the particle are given by”. This question is standard in competitive examination syllabi for Classical Mechanics.