Classical Mechanics MCQ Quiz - Objective Question with Answer for Classical Mechanics - Download Free PDF

Solution:

F₃ = F₃(p, Q, t)

∂F₃/∂p = -q,    ∂F₃/∂Q = -P

⇒ Q = log(1 + q^1/2 cos p) ⇒ e^Q = 1 + q^1/2 cos p

⇒ (e^Q - 1) / cos p = q^1/2 ⇒ q = ((e^Q - 1)²) / cos² p

⇒ P = 2(1 + q^1/2 cos p) q^1/2 sin p ⇒ P = 2e^Q q^1/2 sin p ⇒ 2e^Q(e^Q - 1) tan p

⇒ ∂F₃/∂p = -q = -((e^Q - 1)² / cos² p) ⇒ F₃ = -∫(e^Q - 1)² sec²p dp

⇒ F₃ = - (e^Q - 1)² tan p + f₁(Q)   ........A

∂F₃/∂Q = -P = -2(e^2Q - e^Q) tan p

F₃ = -2 ∫ (e^2Q - e^Q) tan p + f₂(p)

= - (e^Q - 1)² tan p + tan p + f₂(p)   ....(B)

Equating A and B: f₁(Q) = 0,   f₂(p) = -tan p

So F₃ = - (e^Q - 1)² tan p

Solution:

H = ( p – qA )² / 2m + qφ

⇒ H = |p|² / 2m + q² |A|² / 2m – (q / m) p · A

Use p = p_xi + p_yj + p_zk

and A = A_xi + A_yj + A_zk

∂H/∂p_x = (p_x – qA_x) / m = ẋ

∂H/∂p_y = (p_y – qA_y) / m = ẏ

∂H/∂p_z = (p_z – qA_z) / m = ż

Let us find {x, ẏ}:

{x, ẏ} = ∂x/∂x ∂ẏ/∂p_x – ∂x/∂p_x ∂ẏ/∂x + ∂x/∂y ∂ẏ/∂p_y – ∂x/∂p_y ∂ẏ/∂y

0 = – (1/m)(q) ( ∂A_y/∂x – ∂A_x/∂y ) = (q/m²)( ∇ × A )_z = (q/m²) B_k

⇒ {v_i, v_j} = (q/m²) B_k ε_ij ⇒ ε_ijk {v_i, v_j} = (q/m²) B_l ε_ijk ε_ij

= (q/m²) B_l 2δ_kl = 2qB_k / m²

Solution:

L = (15/2) m x² + 6mxẏ + 3my² − mg(x + 2y)

⇒  _{(∂L/∂ẋ) − ∂L/∂x = 0 ⇔ 15mẍ + 6mÿ + mg = 0 ........(1)}

⇒  (∂L/∂ẏ) − ∂L/∂y = 0 ⇔ 6mẍ + 6mÿ + 2mg = 0 .......(2)

Use operation 2(1) − (2)

24mẍ + 6ẏ = 0 ⇒ d/dt (4x + y) = 0 ⇒ 4ẋ + ẏ = 0 ⇒ 12ẋ + 3ẏ = c

The correct answer is Option 1.Key Points

• When an object undergoes acceleration, it means there is a change in its velocity. This change in velocity can occur either in terms of speed, direction, or both.
• A force always acts on it:
  • This statement is generally true.
  • According to Newton's second law of motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma).
  • So, if there's acceleration, there must be a force acting on the object.

Additional Information

• Acceleration is a fundamental concept in physics that describes the rate of change of velocity with respect to time.
• Velocity is a vector quantity, meaning it has both magnitude (speed) and direction. Therefore, any change in speed, direction, or both constitutes acceleration.
• The formula for acceleration (a) is a = F/m where a is acceleration. F is the net force acting on an object and m is the mass of the object.
• This formula states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
• In simpler terms, if you apply a force to an object, it will accelerate, and the acceleration will be larger if the force is stronger or if the object has less mass.
• Acceleration can occur in various forms:
  • Linear Acceleration: Change in speed in a straight line.
  • Angular Acceleration: Change in rotational speed or direction.
  • Centripetal Acceleration: Acceleration directed towards the center of a circular path.
• Acceleration can be positive or negative:
  • Positive Acceleration: Speeding up in the positive direction.
  • Negative Acceleration (Deceleration): Slowing down or moving in the opposite direction.
• Gravity is a common force causing acceleration. Near the Earth's surface, objects in free fall experience acceleration due to gravity, denoted as g (approximately 9.8 m/s²).

Concept:

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.

Calculation:

v =  and v = e

0 = (ev)² - 2gh₁

h₁ =  = e²h

Similarly, h₂ = e⁴h

H = h + 2h₁ + 2h₂ +...∞ 

= h + 2(e²h + e⁴h + ... ∞)

= h + 2e²h()

= h × ()

The coefficient of restitution between the ball and the floor is 0.5.

e = 0.5

H = 5h/3

The correct answer is option (2).

Concept:

We are using the angular momentum formula which is  and then using this formula for  planes.

By using matrices for values of  and  we get the value of magnitude and direction of angular momentum.

Explanation:

 A circular disc is in rotation with the origin as center in  plane.

Given,

•   where  is angular velocity
•  is the moment of inertia

We are using formula for angular momentum

For denoting angular momentum in planes we will use vector notations for angular momentum ,

Using the perpendicular axis theorem,

putting values of  and  in matrix equation, we get

By multiplication above matrices, we get

This is the magnitude of angular momentum. The direction of angular momentum is .

Concept:

 We are using Kepler's law here which states that the radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time.

Explanation:

Using Kepler's second law, 

eccentricity(e)=>

Now,

Explanation:

• Given the transformations  and , we have to check them against the requirements of canonical transformations.
• The main requirement, among others, is that the Poisson bracket {X, P} = 1.
• In terms of partial derivatives, the Poisson bracket can be written as: 
• Substituting X and P from the problem statement, we get: 
• Computing partial derivatives, we get:
• Setting the above equation to 1: 
• Finally, 

Concept:

The Lagranges equation of motion of a system is given by

Calculation:

The Lagrangian L depends on 

L(x,y,,)

L^' = L(x,y,,)

⇒

Similarly 

The correct answer is option (2).

Explanation:

We will first write the velocity vector in polar co-ordinates and write the radial velocity and radial acceleration and by differentiating we get the desired solution.

Given,-     ----------------1

Velocity in polar co-ordinates is given by the sum of radial and transverse velocity.

Radial velocity 

Radial acceleration 

Now,  and 

 => 0\)

 and 

If ,  for some choices of parameters.

So, the correct answer is  -0\) however,  for some choices of parameters.

Concept:

The Hamiltonian of a system specifies its total energy—i.e., the sum of its kinetic energy (that of motion) and its potential energy (that of position)—in terms of the Lagrangian function derived in earlier studies of dynamics and of the position and momentum of each of the particles.

Calculation:

H = q1p1 - q2p2 +  , where a > 0 is a constant.

f = (q₁q₂ + λp₁p₂)

 = [f,H] + 

 = 0 ⇒  = [f,H] = 0

[f,H] = 

q₂ . q₁ - λ p₂ (p₁ + 2aq₁) + q₁(-q₂) - λ p₁(-p₂) = 0

∴ λ = 0

The correct answer is option (1).

Concept:

We will use the relationship of Hamiltonian and Lagrangian which is given by

• For a two-particle system, we can write the above equation as 

Explanation:

Given, 

•  (given)

substitute value of H, we get,

• --------------------------------1
• Using Hamilton equations, 
•  and 
• Put the value of  and  in equation 1, to get the value of Lagrangian
•  

So, the correct answer is .

Concept:

We will first write lagrangian for a given condition then we use the equation for normal nodes which is given by 

Explanation:

Given m are two identical masses, k is spring constant and  and are displacement of spring first and spring second respectively.

• Using matrix we can write operators of  and  as,
•  and 
• Put these values in equation- to get the ratio of normal modes of frequencies,
• Solution of above equation is 
• ratio of both frequencies 

So, the correct answer is