Question

Please solve this question​

Answer 1

$\huge\fcolorbox{black}{aqua}{Solution:-}$

✏️ Question:-

★ Two particles of masses m1 and m2 are located at points defined by position vector r1→ and r2→. forces f1→ and f2→, respectively act on these two particles. using the definition of the centre of mass, show that the product of the total mass (m = m1 + m2) of the given system times the acceleration of its centre of mass equals the vector sum of the forces f1→ and f2→.

✏️ Answer:-

★ Given, the system consists of two masses m1 and m2.

Thus,

Position vectors of the masses m1 and m2 are r1 and r2 respectively.

Therefore,

The position vector of the centre of mass of the system is given by,

$\frac{m1r1 → + m2r2→}{m1 + m2}$

✏️ Explanation:-

★ Center of mass definition:-

• Consider a body consisting of large number of particles whose mass is equal to the total mass of all the particles. When such a body undergoes a translational motion the displacement is produced in each and every particle of the body with respect to their original position.

• If this body is executing motion under the effect of some external forces acting on it then it has been found that there is a point in the system , where whole mass of the system is supposed to be concentrated. The nature the motion executed by the system remains unaltered when force acting on the system is directly applied to this point. Such a point of the system is called center of mass of the system.

• Hence for any system Center of mass is the point where whole mass of the system can be supposed to be concentrated and motion of the system can be defined in terms of the center of mass.

★ Difference between center of mass and center of gravity :-

• The center of mass of a body i a point where whole mass of a body may be assumed to be concentrated for describing its translational motion. On the other hand the center of gravity is the point at which the resultant of the gravitational forces on all the particles of the body acts.

• Note:- For many objects, these two points are in exactly the same place. But they're only the same when the gravitational field is uniform across an object. For example in uniform gravitational field such as that of the earth on a small body , the center of gravity coincides with the center of mass.

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I hope this helps! :)