Classical Mechanics II: Rotational Motion, Gravitation, Oscillations, Waves, and Special Relativity

Question 1

What is a limitation of the Lagrangian formulation in classical mechanics?

Accepted Answer

It is not applicable to problems involving non-conservative forces.

Answer

It is more general than the Newtonian formulation.

Answer

It can be used to solve problems involving constraints.

Answer

It can provide deeper insights into the underlying physics of a system.

Question 2

In classical mechanics, which quantity is NOT conserved within a closed system?

Accepted Answer

Entropy

Answer

Energy

Answer

Momentum

Answer

Angular Momentum

Question 3

When a particle moves in a central force field, which physical quantity remains conserved?

Accepted Answer

Angular momentum

Answer

Linear momentum

Answer

Energy

Answer

Kinetic energy

Question 4

In Hamiltonian mechanics, the momentum coordinate that pairs with the position coordinate is best described as:

Accepted Answer

The rate of change of the Hamiltonian with respect to the position coordinate

Answer

The position coordinate itself

Answer

The total energy of the system

Answer

The kinetic energy of the system

Question 5

Which of the following is NOT a fundamental conservation law in classical mechanics?

Accepted Answer

Conservation of charge

Answer

Conservation of energy

Answer

Conservation of momentum

Answer

Conservation of angular momentum

Question 6

Which quantity is NOT conserved in classical mechanics?

Accepted Answer

Charge

Answer

Energy

Answer

Momentum

Answer

Angular momentum

Question 7

In classical mechanics, which quantity is NOT conserved?

Accepted Answer

Entropy

Answer

Energy

Answer

Momentum

Answer

Angular momentum

Question 8

Which quantity is NOT conserved in classical mechanics?

Accepted Answer

Temperature

Answer

Energy

Answer

Angular momentum

Answer

Momentum

Question 9

Which of the following is not a characteristic of a canonical transformation?

Accepted Answer

Changes the phase space

Answer

Preserves Poisson brackets

Answer

Generated by a function

Answer

Invertible

Question 10

In classical mechanics, which of the following is not a fundamental conserved quantity?

Accepted Answer

Temperature

Answer

Energy

Answer

Linear momentum

Answer

Angular momentum

Question 11

Which of the following is **not** a conserved quantity according to Noether's theorem?

Accepted Answer

Parity

Answer

Energy

Answer

Momentum

Answer

Angular momentum

Question 12

In Hamiltonian mechanics, which transformation is NOT considered canonical?

Accepted Answer

Changes units of coordinates and momenta

Answer

Preserves the Poisson bracket

Answer

Invertible

Answer

Symplectic

Question 13

In classical mechanics, which quantity is **not** conserved?

Accepted Answer

Entropy

Answer

Energy

Answer

Momentum

Answer

Angular Momentum

Question 14

A particle of mass **m** moves in a circular path with radius **R** at a constant speed of **v**. Calculate the particle's angular momentum:

Accepted Answer

mvr

Answer

2mvr

Answer

mv^2/R

Answer

Rmv

Question 15

Calculate the center of mass of a system comprising two particles with masses m1 and m2, positioned at r1 and r2, respectively.

Accepted Answer

The weighted average of the particle positions, with the particle masses as weights

Answer

The midpoint of the line segment connecting the two particles

Answer

The center of the circle that passes through both particles

Question 16

Which of the following is **not** a defining characteristic of canonical transformations?

Accepted Answer

Preservation of energy

Answer

Preservation of Poisson brackets

Answer

Preservation of the symplectic structure

Answer

Preservation of phase space volume

Question 17

To fully describe the orientation of a rigid body relative to a fixed reference frame, how many independent Euler angles are needed?

Accepted Answer

3

Answer

1

Answer

2

Question 18

In classical mechanics, the Hamiltonian formulation uses phase space to describe a system's state. Which quantity is the canonical momentum conjugate to the generalized coordinate q?

Accepted Answer

Velocity $dot{q}$

Answer

Force F

Answer

Potential energy V

Answer

Lagrangian L

Question 19

In classical mechanics, which of the following is NOT a conserved quantity?

Accepted Answer

Velocity

Answer

Total energy

Answer

Total momentum

Answer

Total angular momentum

Question 20

For a particle of mass m moving in a potential V(x), the Lagrangian is given by L = 1/2 mv^2 - V(x). Derive the equation of motion for the particle using the Euler-Lagrange equation.

Accepted Answer

d/dt (dL/dq) - dL/dp = 0

Answer

F = dp/dt

Answer

L = constant

Answer

v = dL/dp

Question 21

In classical mechanics, what geometric object represents the set of all possible states of a system in phase space?

Accepted Answer

Multi-dimensional manifold

Answer

Plane

Answer

Sphere

Question 22

A rigid body rolls without slipping on a horizontal surface. What is the velocity of the point of contact, P, when the angular velocity of the body is ω?

Accepted Answer

ωR

Answer

0

Answer

2ωR

Answer

ωR/2

Question 23

A rigid body rotates with a constant angular velocity about a fixed axis. Which quantity is conserved?

Accepted Answer

Angular Momentum

Answer

Kinetic Energy

Answer

Moment of Inertia

Answer

Angular Velocity

Question 24

In Hamiltonian mechanics, the symplectic structure is crucial for understanding phase space dynamics. Which of the following best describes its significance?

Accepted Answer

Preserves phase space volume as a trajectory evolves.

Answer

Defines the area of infinitesimal surface elements in phase space.

Answer

Determines the curvature of phase space.

Question 25

A spaceship travels at a velocity of 0.8c relative to Earth. An observer on Earth measures the length of the spaceship. What will they observe?

Accepted Answer

Shorter than its proper length

Answer

Longer than its proper length

Answer

Equal to its proper length

Answer

Cannot be determined without additional information

Question 26

Define the angular momentum of a rigid body about a fixed axis.

Accepted Answer

Moment of inertia x angular velocity

Answer

Mass x angular velocity

Answer

Mass x velocity

Answer

Moment of inertia x velocity

Question 27

A simple pendulum consists of a mass 'm' suspended by a string of length 'l'. The pendulum oscillates with an angle 'θ' and a period 'T'. What is the pendulum's period of oscillation?

Accepted Answer

2π√(l / g)

Answer

2π√(g / l)

Answer

π√(l / g)

Answer

π√(g / l)

Question 28

A wave traveling in a medium is described by the equation y = A sin(ωt - kx). What does 'k' represent in this equation?

Accepted Answer

Wave vector

Answer

Angular frequency

Answer

Amplitude

Answer

Wavelength

Question 29

According to special relativity, what is the relationship between an object's mass 'm' and its energy 'E'?

Accepted Answer

E = m * c^2

Answer

E = m / c^2

Answer

E = 2m * c^2

Answer

E = 1 / 2m * c^2

Question 30

Which condition is NOT essential for simple harmonic motion?

Accepted Answer

Oscillating object moving at a constant velocity

Answer

Restoring force proportional to displacement from equilibrium

Answer

System possessing a natural frequency of oscillation

Question 31

Which of the following is NOT a simple harmonic motion?

Accepted Answer

Motion of a comet around the sun

Answer

Motion of a simple pendulum

Answer

Vibration of a guitar string

Answer

Motion of a mass-spring system

Question 32

According to special relativity, an observer moving at a significant fraction of the speed of light will experience:

Accepted Answer

Time dilation and length contraction

Answer

Length dilation and time contraction

Answer

Neither time dilation nor length contraction

Answer

Both time dilation and length dilation

Question 33

A satellite in circular orbit around the Earth experiences a gravitational force:

Accepted Answer

Always directed towards the center of the Earth

Answer

Always perpendicular to its velocity

Answer

Always parallel to its velocity

Answer

Always directed away from the center of the Earth

Question 34

Which of the following is a characteristic of a damped harmonic oscillator?

Accepted Answer

Decreasing amplitude

Answer

Constant amplitude

Answer

Increasing amplitude

Answer

Zero amplitude

Question 35

The Doppler effect for waves is primarily caused by:

Accepted Answer

Change in wave frequency

Answer

Change in wave speed

Answer

Change in wave amplitude

Answer

Change in wave direction

Question 36

In special relativity, the Lorentz factor is given by:

Accepted Answer

√(1 + (v/c)^2)

Answer

√(1 - (v/c)^2)

Answer

1 - (v/c)

Answer

1 + (v/c)

Question 37

A particle moves with a constant velocity 'v' in a circular path. According to special relativity, how does its mass 'm' change compared to its rest mass 'm0'?

Accepted Answer

m = m0√(1 - v^2/c^2)

Answer

m = m0

Answer

m > m0

Answer

m < m0

Question 38

A rigid body rotates about a fixed axis. Which of the following quantities is not conserved during its rotation?

Accepted Answer

Linear momentum

Answer

Angular momentum

Answer

Kinetic energy

Answer

Total energy

Question 39

A harmonic oscillator is driven by a periodic force with a frequency slightly below its natural frequency. How does the amplitude of the oscillator's motion change over time?

Accepted Answer

Decreases

Answer

Increases

Answer

Remains constant

Answer

Oscillates irregularly

Question 40

Which of the following is a vector quantity?

Accepted Answer

Velocity

Answer

Mass

Answer

Temperature

Answer

Time

Question 41

Which of the following is NOT a general property shared by all types of waves?

Accepted Answer

Polarization

Answer

Superposition

Answer

Energy transfer

Answer

Diffraction

Question 42

The angular momentum of a rotating body about its axis of rotation is given by:

Accepted Answer

Moment of inertia times angular velocity

Answer

Mass times velocity times radius

Answer

Force times radius times time

Answer

Torque times angular displacement

Question 43

Which of the following physical quantities is a scalar?

Accepted Answer

Displacement

Answer

Velocity

Answer

Acceleration

Answer

Angular momentum

Question 44

The frequency of a damped oscillator is:

Accepted Answer

Less than the undamped natural frequency

Answer

Equal to the undamped natural frequency

Answer

Greater than the undamped natural frequency

Answer

Dependent on the damping coefficient

Question 45

In rotational dynamics, how does torque relate to angular momentum and angular velocity?

Accepted Answer

Torque is directly proportional to the rate of change of angular momentum.

Answer

Angular momentum is the product of torque and angular velocity.

Answer

Angular velocity is inversely proportional to the torque applied.

Answer

Torque and angular momentum are independent of each other.

Question 46

How does special relativity resolve the apparent contradiction between the constant speed of light in all reference frames and the principle of relativity?

Accepted Answer

Introduces time dilation and length contraction

Answer

Assumes a privileged reference frame

Answer

Rejects the principle of relativity

Answer

Introduces the concept of absolute time

Question 47

A rigid body is rotating about a fixed axis. If its angular velocity is increased by 50%, its kinetic energy will increase by:

Accepted Answer

125%

Answer

50%

Answer

25%

Question 48

A particle of mass m moves in a circular path of radius r with constant speed v. The force acting on the particle is:

Accepted Answer

mv^2/r towards the center of the circle

Answer

mv^2/r away from the center of the circle

Answer

zero

Answer

mg

Question 49

In the special theory of relativity, the Lorentz factor γ for an object moving at a velocity v relative to a stationary observer is:

Accepted Answer

γ = 1/√(1 - v^2/c^2)

Answer

γ = 1 - v/c

Answer

γ = c/v

Answer

γ = v/c

Question 50

The Doppler effect for waves is described by:

Accepted Answer

f' = f(v ± vs) / (v ± vr)

Answer

f' = f(v ± vr) / (v ± vs)

Answer

f' = f(vr ± vs) / (v ± vr)

Answer

f' = f(vs ± vr) / (v ± vr)

Question 51

Which of the following is NOT a characteristic of gravitational force?

Accepted Answer

It can be repulsive

Answer

It is a conservative force

Answer

It is proportional to the product of the masses of the interacting objects

Answer

It acts instantaneously over large distances

Question 52

A simple harmonic oscillator has a mass of 0.5 kg and a spring constant of 100 N/m. What is its natural frequency of oscillation?

Accepted Answer

2 Hz

Answer

4 Hz

Answer

1 Hz

Answer

3 Hz

Question 53

According to special relativity, time

Accepted Answer

Dilates in moving frames of reference

Answer

Is independent of the observer's frame of reference

Answer

Contracts in moving frames of reference

Answer

Passes at a constant rate for all observers

Question 54

A particle moves in a circle of radius R with a constant speed v. Its angular momentum is

Accepted Answer

mvr

Answer

mvr^2

Answer

1/2mvr^2

Answer

2mvr

Question 55

The speed of light in vacuum is approximately

Accepted Answer

3 x 10^8 m/s

Answer

3 x 10^10 m/s

Answer

3 x 10^6 m/s

Answer

3 x 10^4 m/s

Question 56

Which of the following is a vector quantity?

Accepted Answer

Force

Answer

Time

Answer

Mass

Question 57

A wave propagates through a medium with a velocity 'v' and a wavelength 'λ'. The frequency of the wave is:

Accepted Answer

v/λ

Answer

v + λ

Answer

v - λ

Answer

λ/v

Question 58

Within the framework of special relativity, the Lorentz transformation relates the coordinates of an event in one inertial frame to another inertial frame moving at a velocity 'v' relative to the first. Which of the following is a consequence of the Lorentz transformation?

Accepted Answer

Time dilation and length contraction

Answer

Conservation of momentum only

Answer

No change in time or length

Answer

Conservation of energy only

Question 59

A particle of momentum 'p' encounters a potential energy barrier of height 'V'. The de Broglie wavelength of the particle is:

Accepted Answer

h/p

Answer

p/h

Answer

2p/h

Answer

h/2p

Question 60

A solid cylinder of mass 'M' and radius 'R' rolls without slipping on a horizontal surface. The kinetic energy of the cylinder is:

Accepted Answer

(1/2)Mv² + (1/4)MR²ω²

Answer

Zero

Answer

Mv²

Answer

MR²ω²

Question 61

A satellite of mass 'm' orbits Earth (mass 'M') in a circular path of radius 'r'. The potential energy of the satellite is:

Accepted Answer

-GmM/r

Answer

GmM/r

Answer

-GmM/2r

Answer

GmM/2r

Question 62

A spring-mass system undergoes simple harmonic motion. The total mechanical energy of the system is:

Accepted Answer

Constant

Answer

Increasing

Answer

Zero

Answer

Decreasing

Question 63

Which of the following types of equilibrium is NOT exhibited by a rigid body?

Accepted Answer

Dynamic equilibrium

Answer

Unstable equilibrium

Answer

Neutral equilibrium

Answer

Stable equilibrium

Question 64

Identify the fundamental postulate of special relativity.

Accepted Answer

The speed of light is constant in all inertial frames of reference

Answer

Length is absolute

Answer

Time is absolute