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**Centre of Mass**

Consider a system composed of N particles with each particle's mass described by m_{i}, where i is an index from i = 1 to i = N . The total mass of the system is denoted by M,

where the summation over i runs from i = 1 to i = N.

Such a system is displayed in figure given. If the vector connecting the origin with the i^{th} particle is r_{i} then the vector defining the position of the system's center of mass is

The position of the centre of mass is defined by R_{CM} ≡

For N number of mass elements, if r_{j }is the position of the j^{th} element, and m_{j} is its mass, then the center of mass is defined as

**Example 1: The position vector of three particles of mass m _{1 }= 1kg, m_{2} = 2 kg and m_{3} = 3kg are **

**respectively. Find the position vector of their centre of mass.**

The position vector of COM of the three particles will be given by

Substituting the values, we get

**Example 2: Four particles of mass 1 kg, 2kg, 3kg and 4kg are placed at the four vertices A,B, C and D of a square of side 1 m. Find the position of centre of mass of the particles.**

Assuming D as the origin, DC as x - axis and DA as y - axis, we have

m_{1}= 1kg, x_{1}, y_{1}) = ( 0,1 m)

m_{2}= 2kg, x_{2}, y_{2}) = (1m,1m)

m_{3}= 3kg, x_{3}, y_{3}) = (1 m, 0)

m_{4}_{ }= 4kg, x_{4}, y_{4}) = (0, 0)

and co-ordinates of their COM are

Similarly,

∴ (x_{COM}y_{COM = }(0.5,0.3)

**Rigid Body Dynamics**

A rigid body is defined as system of particles in which the distance between any two particles remains fixed throughout the motion.

**Degree of Freedom of Rigid Body**

To define rigid body, there must be minimum 3 non-collinear points.

Let P_{1} (x_{1}, y_{1}, z_{1}), P_{2} (x_{2}, y_{2} , z_{2}) , P_{3} (x_{3}, y_{3}, z_{3}) are three non-collinear points.

So, equation of constrained is

r_{12 }= (x_{1} - x_{2})2 + (y_{1} - y_{2})^{2} + (z_{1} - z_{2})^{2} = c_{1}

r_{23} = (x_{2} - x_{3})_{2} + (y_{2} - y_{3})^{2} + (z_{2} - z_{3})^{2} = c_{2}

r_{13} = (x_{1} - x_{3})_{2} + (y_{1} - y_{3})^{2} + (z_{1} - z_{3})^{2} = c_{3}

Dof = 3N - K

3 x 3 - 3 = 6

So there is six degree of freedom for rigid body

**Center of Mass of Continuous System**

The result is not rigorous, since the mass elements are not true particles. However, in the limit where N approaches infinity, the size of each element approaches zero and the approximation becomes exact.

This limiting process defines an integral.

Formallywhere dm is a differential mass element. Then,

To visualize this integral, think of dm as the mass in an element of volume dV located at position r. If the mass density at the element is ρ, then dm = ρdV and . This

integral is called a volume integral.**Example 3: A rod of length L is placed along the x - axis between x = 0 and x = L. The linear density (mass/length) λ of the rod varies with the distance x from the origin as λ = Rx . Here, R is a positive constant. Find the position of centre of mass of this rod.**

Mass of element dx situated at x = x is

dm = λdx = Rx dx

The COM of the element has coordinates (x,0,0).

Therefore, x -coordinate of COM of the rod will be

The y -coordinate of COM of the rod is y_{COM }(as y = 0 )

Similarly, z_{COM}= 0 Hence, the centre of mass of the rod lies at

**Example 4: Find the center of mass of a thin rectangular plate with sides of length α and b, whose mass per unit area σ varies in the following fashion:σ = σ0(xy/αb) where σ _{0} is a constant.**

We find M , the mass of the plate, as follows:

We first integrate over x , treating y as a constant.

The x component of R is

Similarly,

⇒ y = 2/3 b So center of mass is

**Example 5 : Find the centre of mass of semicircular disc of mass M and radius R.**

From the circular symmetry x = r cos θ, y = r sin θ

**Example 6: Find the centre of mass of a hollow hemisphere of radius R and mass M.**

Consider a ring at angle θ from the base area of ring This problem can be solved in spherical symmetry

x = R sinθ cos∅ , y = R sinθ sin∅ , z = R cos θ

X_{C.M}= ∫xdm

R sin θ cos ∅.sin θdθd∅

**Second method**

The integral is over three dimensions, but the symmetry of the situations lets us treat it as a one dimensional integral. We mentally subdivide the hemisphere into a pile of thin disks. Consider the circular disk of radius r and thickness dz . Its volume is dV = πr^{2}dz , and its mass is

dM = ρdV= (M/V) dV, where V = 2/3 πR^{3}.

Hence,

To evaluate the integral, we need to find r in terms of z. Since, r^{2} = R^{2 }- z^{2}, we have

**Example 7: Find the centre of mass triangular plate of Mass M . Where l and h are given parameter shown in figure.**

lf some mass M_{1}is removed from a rigid body of mass M , then the position of centre of mass of the remaining portion is obtained from the following formulae:Here, the values mass and center of mass of the whole mass while arethe values for the mass which has been removed.

**Example 8: Find the position of centre of mass of the remaining plate when a circle is cut from right half radius of circle is l/8. Centre of circle is in centre of the right part of the plate.**

Area

A_{1}l^{2}/4 l/4 l/2

A_{2}l^{2}/4 3l/8 l/2

A_{3}πl^{2}/64 3l/8 l/2

Taking origin at corner O

Centre of mass is (0.28l, 0.5l).

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