Classical Mechanics MCQ Quiz - Objective Question with Answer for Classical Mechanics - Download Free PDF
Last updated on Jun 28, 2025
Latest Classical Mechanics MCQ Objective Questions
Classical Mechanics Question 1:
For the transformation
Q=ln (1+𝑞1/2 cos 𝑝) ,𝑃 = 2𝑞 1/2 (1+𝑞1/2cos 𝑝) sin 𝑝
the generating function is
Answer (Detailed Solution Below)
Classical Mechanics Question 1 Detailed Solution
Solution:
F3 = F3(p, Q, t)
∂F3/∂p = -q, ∂F3/∂Q = -P
⇒ Q = log(1 + q1/2 cos p) ⇒ eQ = 1 + q1/2 cos p
⇒ (eQ - 1) / cos p = q1/2 ⇒ q = ((eQ - 1)2) / cos2 p
⇒ P = 2(1 + q1/2 cos p) q1/2 sin p ⇒ P = 2eQ q1/2 sin p ⇒ 2eQ(eQ - 1) tan p
⇒ ∂F3/∂p = -q = -((eQ - 1)2 / cos2 p) ⇒ F3 = -∫(eQ - 1)2 sec2p dp
⇒ F3 = - (eQ - 1)2 tan p + f1(Q) ........A
∂F3/∂Q = -P = -2(e2Q - eQ) tan p
F3 = -2 ∫ (e2Q - eQ) tan p + f2(p)
= - (eQ - 1)2 tan p + tan p + f2(p) ....(B)
Equating A and B: f1(Q) = 0, f2(p) = -tan p
So F3 = - (eQ - 1)2 tan p
Classical Mechanics Question 2:
A non-relativistic particle of mass 𝑚 and charge 𝑞 is moving in a magnetic field 𝐵⃗(𝑥,𝑦,𝑧). If 𝑣⃗denotes its velocity and {…}P.B. denotes the Poisson Bracket, then
Answer (Detailed Solution Below)
Classical Mechanics Question 2 Detailed Solution
Solution:
H = ( p – qA )2 / 2m + qφ
⇒ H = |p|2 / 2m + q2 |A|2 / 2m – (q / m) p · A
Use p = pxi + pyj + pzk
and A = Axi + Ayj + Azk
∂H/∂px = (px – qAx) / m = ẋ
∂H/∂py = (py – qAy) / m = ẏ
∂H/∂pz = (pz – qAz) / m = ż
Let us find {x, ẏ}:
{x, ẏ} = ∂x/∂x ∂ẏ/∂px – ∂x/∂px ∂ẏ/∂x + ∂x/∂y ∂ẏ/∂py – ∂x/∂py ∂ẏ/∂y
0 = – (1/m)(q) ( ∂Ay/∂x – ∂Ax/∂y ) = (q/m2)( ∇ × A )z = (q/m2) Bk
⇒ {vi, vj} = (q/m2) Bk εij ⇒ εijk {vi, vj} = (q/m2) Bl εijk εij
= (q/m2) Bl 2δkl = 2qBk / m2
Classical Mechanics Question 3:
The Lagrangian of a system is
Which one of the following is conserved?
Answer (Detailed Solution Below)
Classical Mechanics Question 3 Detailed Solution
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
Classical Mechanics Question 4:
A frictionless track is defined by 𝑧 = 𝑧o -
A particle is constrained to slide down the track under the action of gravity. The tangential acceleration at position (𝑥,𝑧) would be
Answer (Detailed Solution Below)
Classical Mechanics Question 4 Detailed Solution
Calculation:
Tangential acceleration is given by:
aθ = g ⋅ sinθ
z = z0 - (x2 / 4z0)
tanθ = (dz / dx) = - (x / 2z0) ⇒ sinθ = (x / √(x2 + 4z02)) ⇒ aθ = (g ⋅ x) / √(x2 + 4z02)
Classical Mechanics Question 5:
A particle of rest mass 𝑚o and energy 𝐸 collides with another particle at rest, with the same rest mass. What is the minimum value of 𝐸 so that after the collision, there may be four particles of rest mass 𝑚o ?
Answer (Detailed Solution Below)
Classical Mechanics Question 5 Detailed Solution
Solution:
For the first particle, the four-vector momentum before collision is:
( E / c , p, 0, 0 )
Second particle is at rest, so its momentum vector before collision is:
( m0c2 / c , 0, 0, 0 )
Where,
p2 = (E2 / c2) - m02 c2
The quantity:
∑i (E2 / c2 - p2)
is invariant before and after the collision.
Therefore,
((E + m0c2) / c)2 - p2 ≥ (4m0c2)2 / c2
⇒ ((E + m0c2) / c)2 - [(E2 / c2) - m02 c2] ≥ (4m0c2)2 / c2
For minimum energy,
((E + m0c2) / c)2 - [(E2 / c2) - m02 c2] = (4m0c2)2 / c2
⇒ E = 7 m0c2
Top Classical Mechanics MCQ Objective Questions
When an object undergoes acceleration
Answer (Detailed Solution Below)
Classical Mechanics Question 6 Detailed Solution
Download Solution PDFThe 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).
A ball, initially at rest, is dropped from a height h above the floor bounces again and again vertically. If the coefficient of restitution between the ball and the floor is 0.5, the total distance travelled by the ball before it comes to rest is
Answer (Detailed Solution Below)
Classical Mechanics Question 7 Detailed Solution
Download Solution PDFConcept:
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 =
0 = (ev)2 - 2gh1
h1 =
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).
A uniform circular disc on the xy-plane with its centre at the origin has a moment of inertia I0 about the x- axis. If the disc is set in rotation about the origin with an angular velocity ω = ω0(ĵ + k̂), the direction of its angular momentum is along
Answer (Detailed Solution Below)
Classical Mechanics Question 8 Detailed Solution
Download Solution PDFConcept:
We are using the angular momentum formula which is
By using matrices for values of
Explanation:
A circular disc is in rotation with the origin as center in
Given,
where is angular velocity is the moment of inertia
We are using formula for angular momentum
For denoting angular momentum in
Using the perpendicular axis theorem,
putting values of
By multiplication above matrices, we get
This is the magnitude of angular momentum. The direction of angular momentum is
The minor axis of Earth's elliptical orbit divides the area within it into two halves. The eccentricity of the orbit is 0.0167. The difference in time spent by Earth in the two halves is closest to
Answer (Detailed Solution Below)
Classical Mechanics Question 9 Detailed Solution
Download Solution PDFConcept:
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,
For the transformation x → X =
Answer (Detailed Solution Below)
Classical Mechanics Question 10 Detailed Solution
Download Solution PDFExplanation:
- 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,
Which of the following terms, when added to the Lagrangian L(x, y,
Answer (Detailed Solution Below)
Classical Mechanics Question 11 Detailed Solution
Download Solution PDFConcept:
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).
The trajectory of a particle moving in a plane is expressed in polar coordinates (r, θ) by the equations
Answer (Detailed Solution Below)
Classical Mechanics Question 12 Detailed Solution
Download Solution PDFExplanation:
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,-
Velocity in polar co-ordinates is given by the sum of radial and transverse velocity.
Radial velocity
Radial acceleration
Now,
If
So, the correct answer is -
The Hamiltonian of a system with two degrees of freedom is H = q1p1 - q2p2 +
Answer (Detailed Solution Below)
Classical Mechanics Question 13 Detailed Solution
Download Solution PDFConcept:
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 +
f = (q1q2 + λp1p2)
[f,H] =
q2 . q1 - λ p2 (p1 + 2aq1) + q1(-q2) - λ p1(-p2) = 0
∴ λ = 0
The correct answer is option (1).
The Hamiltonian of a two particle system is H = p1p2 + q1q2, where q1 and q2 are generalized coordinates and p1 and p2 are the respective canonical momenta. The Lagrangian of this system is
Answer (Detailed Solution Below)
Classical Mechanics Question 14 Detailed Solution
Download Solution PDFConcept:
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
A system of two identical masses connected by identical springs, as shown in the figure, oscillates along the vertical direction.
The ratio of the frequencies of the normal modes is
Answer (Detailed Solution Below)
Classical Mechanics Question 15 Detailed Solution
Download Solution PDFConcept:
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
- 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