# The linear momentum of a system of two particles is expressed as 20î + 30ĵ + 40k̂ and 20î + 40ĵ + 60k̂ units respectively. If their centre of mass

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The linear momentum of a system of two particles is expressed as 20î + 30ĵ + 40k̂ and 20î + 40ĵ + 60k̂ units respectively. If their centre of mass has a velocity of 4î + 5ĵ + 7k̂ unit, the mass of the system is-
1. 5 units
2. 10 units
3. 15 units
4. 20 units

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Correct Answer - Option 2 : 10 units

The correct answer is option 2) i.e. 10 units

CONCEPT:

• Linear momentum: Momentum is defined as the impact due to a moving object.
• Linear momentum, p = mv

Where m is the mass of the object and v is the velocity of the object.

• Linear momentum of a system of particles:

Let us consider a system of particles with each particle having a mass m and velocity v.

If p is the linear momentum of each particle, p = mv

Total linear momentum, Σp = Σmv = Σm . Σv = M.vcom

Where M is the sum of the mass of all particles, and vcom is the velocity of the centre of mass.

CALCULATION:

Given that:

p1 = 20î + 30ĵ + 40k̂ and p2 = 20î + 40ĵ + 60k̂

Velocity of the centre of mass, vcom = 4î + 5ĵ + 7k̂

Let M be the mass of the system.

We know that,

Σp = Mvcom

⇒ (20î + 30ĵ + 40k̂) + (20î + 40ĵ + 60k̂) = M (4î + 5ĵ + 7k̂)

Considering the x direction,

⇒ (20î) + (20î) = M(4î)    (Mass is a scalar quantity.It will have no components in different directions)

⇒ 40î = M 4î

⇒ M = 10 units