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Two objects of masses $200g$ and $500g$ possess velocities $10\hat im/s$ and $3\hat i + 5\hat j$ m/s respectively. The velocity of their centre of mass in m/s is
a. $5\hat i - 25\hat j$
b. $\dfrac{5}{7}\hat i - 25\hat j$
c. $5\hat i - \dfrac{{25}}{7}\hat j$
d. $25\hat i - \dfrac{5}{7}\hat j$

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Last updated date: 13th Jun 2024
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Answer
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Hint: To describe its motion, we consider a point in the body where the entire mass of the body is supposed to concentrate to describe its motion is called center of mass. Motion of the body is represented by the path of the particle at the center of mass point. Using the above data apply the velocity of mass formula.

Complete step by step answer:
The total momentum of the body is conserved when the initial momentum is equal to the final momentum of a system.
To describe its motion, we consider a point in the body where the entire mass of the body is supposed to concentrate to describe its motion is called center of mass.
Motion of the body is represented by the path of the particle at the center of mass point.
The center of mass is located at the centroid when the rigid body is with uniform density. The center of mass for a disc which is uniform would be at a center.
In some cases, the center of mass may not fall on the object. For a ring the center of mass is located at its center.

Let us consider two blocks A and B
${M_A} = 200g = 0.2kg$ and ${M_B} = 500g = 0.5kg$
Initial momentum is given by
$ \Rightarrow {M_A}{V_A} + {M_B}{V_B} = 0.2 \times 10\hat i + 0.5 \times \left( {3\hat i + 5\hat j} \right)$
$ \Rightarrow {M_i} = 3.5\hat i + 2.5\hat j$
Then the final momentum
$ \Rightarrow {M_{total}}{V_{cm}} = 0.7{V_{cm}}$
Then from conservation of momentum,
$ \Rightarrow {M_i} = 3.5i + 2.5j = 0.7{V_{cm}}$
Hence velocity of center of mass,
\[ \Rightarrow {V_{cm}} = 5\hat i + \dfrac{{25}}{7}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} \]

Hence, the correct answer is option (C).

Note: By vector addition, we can determine the center of mass of an object. If the particle moves in uniform velocity then the magnitude of the center of mass is obtained by parallelogram law of vectors. The center of mass is located at the centroid when the rigid body is with uniform density. The center of mass for a disc which is uniform would be at a center.