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Question :
If two particles of mass m₁ = 2kg and m₃ = 3kg have position vectors
- r₁ = t i + 2t² j + 2t k
- r₂ = i + 2t³ j + 5 k
The position vectors of r₁ and r₂ are in metres and time is in seconds. Calculate the velocity and acceleration of centre of mass of the two particles system
Solution :
❖ Centre of mass is an imaginary point where the whole mass of system can be assumed to be concentrated.
A] Velocity of first particle :
➙ v₁ = d (r₁) / dt
➙ v₁ = d (t i + 2t² j + 2t k) / dt
➙ v₁ = i + 4t j + 2 k m/s
B] Velocity of second particle :
➙ v₂ = d (r₂) / dt
➙ v₂ = d (i + 2t³ j + 5 k) / dt
➙ v₂ = 6t² j + k m/s
♦ Velocity of centre of mass :
➠ v = (m₁v₁ + m₂v₂) / (m₁ + m₂)
➠ v = [2(i + 4t j + 2 k) + 3(6t² j + k)] / 5
➠ v = [2 i+ 8t j + 4 k + 18t² j + 3 k] / 5
➠ v = [2 i + 2t(4 + 9t) j + 7 k] / 5 m/s
♦ Acceleration of centre of mass :
➠ a = dv / dt
➠ a = d [(2 i + 8t j + 4 k + 18t² j + 3 k) / 5] / dt
➠ a = [(8 j + 36t j) / 5] / dt
➠ a = [4(2 + 9t) j / 5] m/s²
QUESTION:-
If two particles of mass m₁ = 2kg and m₃ = 3kg have position vectors
r₁ = t i + 2t² j + 2t k
r₂ = i + 2t³ j + 5 k
The position vectors of r₁ and r₂ are in metres and time is in seconds. Calculate the velocity and acceleration of centre of mass of the two particles system..
SOLUTION:-
❖ Centre of mass is an imaginary point where the whole mass of system can be assumed to be concentrated.
A] Velocity of first particle :
➙ v₁ = d (r₁) / dt
➙ v₁ = d (t i + 2t² j + 2t k) / dt
➙ v₁ = i + 4t j + 2 k m/s
B] Velocity of second particle :
➙ v₂ = d (r₂) / dt
➙ v₂ = d (i + 2t³ j + 5 k) / dt
➙ v₂ = 6t² j + k m/s
♦ Velocity of ventre of mass:
➠ v = (m₁v₁ + m₂v₂) / (m₁ + m₂)
➠ v = [2(i + 4t j + 2 k) + 3(6t² j + k)] / 5
➠ v = [2 i+ 8t j + 4 k + 18t² j + 3 k] / 5
➠ v = [2 i + 2t(4 + 9t) j + 7 k] / 5 m/s
♦ Acceleration of centre of mass :
➠ a = dv / dt
➠ a = d [(2 i + 8t j + 4 k + 18t² j + 3 k) / 5] / dt
➠ a = [(8 j + 36t j) / 5] / dt
➠ a = [4(2 + 9t) j / 5] m/s²