Center of Mass and Linear Momentum MCQ Quiz - Objective Question with Answer for Center of Mass and Linear Momentum - Download Free PDF

Explanation:

According to Newton's second law, Fext = dp/dt, in the absence of external forces (Fext = 0), the equation simplifies to dp/dt = 0. This means that the linear momentum p of the system is constant when the resultant of all external forces is zero.

The kinetic energy of the system can still change, as seen in inelastic collisions. Similarly, angular momentum may change if an external torque is applied, such as in the case of a couple acting on a rod. Angular momentum is conserved only when there is no external torque.

Thus, the correct answer is (A): The linear momentum of the system does not change in time.

Calculation:

In a perfectly elastic collision:

The relative speed after collision equals the relative speed before collision because the coefficient of restitution e = 1.

This relationship comes from the definition of elasticity, not directly from momentum conservation.

Linear momentum is also conserved in all types of collisions (elastic and inelastic), provided no external force acts.

Hence, both statements are true, but the second is not a correct explanation of the first.

Calculation:

As the ball falls, its gravitational potential energy decreases and kinetic energy increases. When it contacts the surface, the kinetic energy is stored as elastic potential energy (like a spring), forming a peak. After rebounding, the cycle repeats. The variation in potential energy with time is thus smooth and periodic, with parabolic peaks at collision points.

Correct Option: (B)

Concept Used:

When two objects are in free fall (ignoring air resistance), their motion is governed by the force of gravity. The following points are relevant:

Acceleration: All objects experience the same acceleration due to gravity (g = 9.8 m/s²) regardless of their mass.

Momentum: Momentum (p) = mass × velocity. Since the bodies have different masses, their momenta will differ.

Kinetic Energy: KE = (1/2) × mass × velocity². As mass differs, so does kinetic energy.

Potential Energy: PE = mass × g × height. Again, different masses result in different potential energies.

∴ The only quantity equal for both bodies is acceleration.

∴ The correct answer is Option 2: Acceleration.

Concept:

Momentum:

momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction.

• The unit of momentum (P) is kg m/s.
• Dimension: [MLT^-1]

Law of conservation of Momentum: 

A conservation law states that the total linear momentum of a closed system remains constant through time, regardless of other possible changes within the system.

P₁ = P₂

m₁ v₁ = m₂ v₂

Where, P₁ = initial momentum of system, P₂ = final momentum of system, m₁ = mass of first object, v₁ = velocity of first object, m₂ = mass of second object and v₂ = velocity of second object.

Calculation:

Given:  m₁ = 2 kg    m₂ = 3 kg     u₁ = 5 m/s        u₂ ​=  0 m/s

Let the common velocity of the combined body be V m/s

Mass of combined body      M = 2 + 3 = 5 kg

Applying conservation of momentum:          

m₁ v₁ + m₂ v₂ = M V

⇒ (2 × 5) + (3 × 0) = 5 V

⇒ 10 + 0 = 5 V

⇒ V = 2 m/s

Hence the combined velocity of both the spheres is 2 m/s.

Concept:

Second Law of Motion:

• The rate of change of momentum is directly proportional to the applied force. 

⇒ F × t = Δ p

Where F is force, t is time, and Δ p is change in momentum. 

Calculation:

Given: Initial momentum (p_i) = 10 kg m s-1, Force  (F) = 30 N, and time (t) = 3 second

Let final momentum be p_f

• Change in momentum is

⇒ Δ p = F × t

⇒ Δ p = 30 N × 3 s = 90 N s = 90 kg m s^-1

• As we know the change in momentum is equal to the difference between the final momentum and initial momentum i.e., 

⇒ Δ p = pf - pi 

⇒ p_f = Δ p + p_i

⇒ p_f = 90 kg m s^-1 + 10 kg m s^-1 = 100 kg m s^-1

• So, the correct option is 100 kg m s-1

CONCEPT:

• Momentum (P): The product of mass and velocity is called momentum.
• The SI unit of momentum is kg m/s.
• Momentum (P) = Mass (m) × Velocity (v)

EXPLANATION:

Since, P = m v

Let the initial velocity of the body be v

Since the Mass of the body is constant 

So, P₁ = m v      ----(i)

According to the question

The new velocity of the body = 2v 

New Momentum (P₂) = m × 2v

⇒ P2 = 2mv        ----(ii)

On dividing (ii) by (i), we get 

⇒ P₂ = 2 P₁

∴ The momentum will be doubled.

Key Points

• If the velocity of a body is doubled then its momentum doubles because velocity is directly proportional to momentum. So option 2 is correct.

The correct answer is 500 kg.

CONCEPT:

• The type of collision in which there is a loss of kinetic energy and after the collision both the colliding particles move together is called perfectly inelastic collision.
• In this type of collision the coefficient of restitution is equal to 0.
• Momentum of the system remains constant in this collision.

• Momentum before collision (P1) = Momentum after the collision (P2)

P1 = m1u1 + m2u2

P2 = m1v1 + m2v2

CALCULATION:

Here, speed of the car is 0 m/s

⇒ After collision speed is 8 m/s

According to "Principle of conservation of momentum"

⇒ So, 8 = (m₁v₁ + m₂v₂)/(m₁ + m₂) where m₁ is the mass of truck, v₁ is the velocity of truck, v₂ is the velocity of car and m₂ is the mass of car.

⇒ 8 = (2000 × 10 + m₂ × 0)/(2000 + m₂)

⇒ 16000 + 8 m₂ = 20000

Therefore, mass of the car is 500 kg.

CONCEPT:

• Momentum: A property of a body in motion that is equal to the product of the body's mass and velocity is called momentum.

P = mv

where P is the momentum of the body, m is the mass of the body, and v is the velocity of the body.

• Momentum Conservation: When in a system, there is no external force then the total momentum (P) of the system will be conserved.
• Kinetic energy: The energy in a body due to its motion, is known as kinetic energy.

The kinetic energy in terms of momentum is given by:

where K is the kinetic energy of the body, P is the momentum of the body and m is the mass of the body.

CALCULATION:

Let initial value of P = 100, m = 100

After increase of 30% P = 130

Initial kinetic energy 

Final kinetic energy 

% Increase in Kinetic energy = %

So the correct answer is option 3.

CONCEPT:

• In classical mechanics when an object is projected upward with some initial velocity, and as it moves upward its initial velocity keeps on decreasing because of gravitational acceleration since it acted downward.
• At the highest point, the final velocity will become zero and the time taken to reach that point can be calculated using a kinematic equation 

v = u + gt

where v = final velocity, u = initial velocity, g = acceleration due to gravity and t = time

Given,

Distance travelled (s) = 10 m,

Final velocity (v) = 0 m/s,

Acceleration due to gravity (g) = 9.8 m/s², 

Acceleration of the object, a = –9.8 m/s² (upward motion)

• The velocity with which the object was thrown upwards will be

⇒ v² = u² + 2as 

⇒ 0 = u² + 2 × (–9.8 m/s²) × 10 m

⇒ –u² = –2 × 9.8 × 10 m²/s²

⇒ u = 196 m²/s²

⇒ u = 14 m/s

CONCEPT:

• Projectile motion: Projectile motion is the motion of an object projected into the air, under only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.
  • Initial Velocity: The initial velocity can be given as x components and y components.

ux = u cosθ

uy = u sinθ

Where u stands for initial velocity magnitude and θ refers to projectile angle.

• Maximum Height: The maximum height is reached when vy = 0.

Where h is the maximum height.

• Range: The range of the motion is fixed by the condition y = 0.

Where R is the total distance covered by the projectile.

Kinetic energy (KE) = (1/2)m V²

Where m is mass and V is the velocity

CALCULATION:

Given that:

(Angle with vertical = 30°)

θ = 60° (with horizontal 90o - 30o = 60o )

ux = u cosθ = u cos 60° = (1/2) u

uy = u sinθ = u sin 60° =√3/2 / 2

Kinetic energy on the ground (KE) = (1/2)m u2 = K

At the top, velocity = V = u / 2

Kinetic energy at top (KE') = (1/2)m (u/2)2 = mu²/8 = K/4

So option 1 is correct.

CONCEPT:

• Momentum: Momentum is the product of the mass and the velocity of an object or particle. It is a vector quantity.
  • The standard unit of momentum magnitude is the kilogram-meter per second (kg·m/s).

P = m v

Where P = momentum, m = mass of the body, v = velocity of body.

• Conservation of Linear Momentum: Conservation of Linear Momentum states that a body in motion retains its total momentum (product of mass and vector velocity) unless an external force is applied to it.
• In mathematical term

Initial momentum = Final momentum

P₁ = P₂

We can say that

M₁V₁ + M₂V₂ = 0

Where M₁ = mass of projectile, V₁ = velocity of projectile, M₂ = mass of tank and V₂ = Unknown recoil speed of tank

CALCULATION:

Given that M₁ = 3 kg, V₁ = 24 m/s, M₂ = 0.4 tonnes = 400 kg

Substituting these values in the formula:

M₁V₁ + M₂V₂ = 0

(3 × 24) + (400 × V₂) = 0

V₂ = -0.18 m/s,

The tank recoils in the backward direction with a magnitude of 0.18 m/s.

So option 2 is correct.

Concept:

position of the centre of mass:

The velocity of the centre of mass:

Acceleration of the centre of mass:

Calculations:

 (Negative because both are moving in the opposite direction)

= 0

CONCEPT:

• Kinetic energy (K.E): The energy possessed by a body by the virtue of its motion is called kinetic energy.

The expression for kinetic energy is given by:

Where m = mass of the body and v = velocity of the body

• Momentum (p): The product of mass and velocity is called momentum.

Momentum (p) = mass (m) × velocity (v)

The relationship between the kinetic energy and Linear momentum is given by:

As we know,

Divide numerator and denominator by m, we get

 [p = mv]

CALCULATION:

Given that:

K.E1 = K.E2= K.E (let say)

m₁ = 1 kg and m₂ = 4 kg

The relation between the momentum and the kinetic energy is given by:

But as K.E is the same

∴ 

Or, 

So option 1 is correct.