This article contains Center of mass Problems for class 11. In this article I have given numerical problems, very short answer type questions , short answer type questions and some extra questions. 
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Center of mass
Other assignments(Included with Linear momentum) for this chapter which you can have a look are
Assignment 1
Assignment 2
Assignment 3
Assignment 4

Center of mass Problems

Numerical Problems

Question 1 Find the center of mass of a triangular lamina.

Question 2 The masses of 3, 4 and 5 Kg are located at the corners of the equilateral triangle of side 1 m. Locate the center of mass of the system.

Question 3 The particles of masses 100 gm and 300 gm at a given time have positions $2\hat i +5 \hat j + 13 \hat k$ and $-6\hat i +4 \hat j -2 \hat k$ m respectively and velocities $10\hat i -7 \hat j -3 \hat k$ and $7\hat i -9 \hat j + 6 \hat k$ m/s respectively. Determine the instantaneous position and velocity of CM.

Question 4 Show that the center of mass of a uniform rod of mass $M$ and length $L$ lies at the middle point of the rod.

Question 5 Determine the center of mass of a hemisphere of radius $R$.

Question 6 A circular plate of uniform thickness has a diameter 56 cm. A circular portion of 42 cm is removed from one edge of the plate. Find the CM of the remaining portion.

Question 7 The identical spheres A, B and C each of radius R are placed touching each other on a horizontal table. Where is the center of mass of the system located relative to center of mass of the sphere A?

Question 8 Two blocks of mass 2 Kg and 5 Kg are placed on a frictionless table and connected by a spring. On giving a jerk, the heavier block starts moving towards the other block with velocity 9.8 m/s. Find the velocity of the center of mass.

Question 9 A thin rod of length $l$ and mass $m$ has a disk which is attached to one of its ends such that rod and disk are coplanar. Find CM from the center of disk, if mass of the disk is M and radius is $R$.

Question 10 Four particles of masses 2m, m , 4m and 3m are placed at the corners A, B, C and D of a square of each side x as shown below in the figure.

Find the position of the center of mass of the system.

Very Short answer type questions

Question 1 Where does the center of mass of a two particle system lie , if one particle is more massive then another.

Question 2 Is center of mass a reality?

Question 3 Where does center of mass of a rectangle lie?

Question 4 Should center of mass of a body necessarily lie inside the body?

Question 5 What will be nature of motion of center of mass of an isolated system?

Question 6 How is the center of mass depends on the relative distance between the particles?

Question 7 Write two factors on which center of mass of a body does not depend.

Question 8 Give the location of a center of mass of a ring? does it it lie inside the body?

Question 9 Define motion of center of mass of a fire cracker that explode in air before and after exposition?

Short Answer type questions

Question 1 A body A of mass M while falling vertically downwards under the gravity breaks into two parts; a body B of mass M/3 and a body C of mass 2M/3. How does the center of mass of bodies B and C taken together shift compared to that of body A?

Question 2 Prove that the center of mass of two particles divides the line joining the particles in the inverse ratio of their masses.

Question 3 Distinguish between the center of gravity of a body and its center of mass.

Question 4 Two identical particles move towards each other with velocity $2v$ and $v$ respectively. What is the velocity of the center of mass?

Question 5 What is the significance of defining center of mass of a system?

Extra Questions

Question 1 Obtain an expression for the position vector of the center of mass of a system containing two particles.

Question 2 Discuss the motion of center of mass in the following systems

(a) Earth moon system revolving around the sun

(b) Diatomic molecule

Question 3 What do you mean by center of mass of a system. Prove that center of mass of a system moves with constant velocity in the absence of external force on the system.

Answers to selected Center of mass Problems

Question 1

CM lies on the point of concurrence of the medians i.e., on the centroid G of the triangle.

Question 2

The coordinates of CM are (0.54 m, 0.36 m)

Question 3

${{\vec{R}}_{CM}}=\frac{-16\hat{i}+17\hat{j}+7\hat{k}}{4}m$

${{\vec{v}}_{CM}}=\frac{31\hat{i}-34\hat{j}+15\hat{k}}{4}m{{s}^{-1}}$

Question 6

9 cm

Question 7

$\frac{1}{3}(\vec{AB}+\vec{AC})$

Question 8

7 ms^-1

Question 9

${{X}_{CM}}=\frac{m(2r+l)}{2(M+m)}$

Question 10

(0.5x , 0.7x)

Some other resources you can look for reference are
https://en.wikipedia.org/wiki/Center_of_mass
https://www.khanacademy.org/science/physics/linear-momentum/center-of-mass/a/what-is-center-of-mass
http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html
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