Question

Half of the rectangular plate made of a material of density d1 and next half of it made of density d2 .The length of the plate is L.Locate the centre of mass of the plate.

Answer 1

The centre of mass of the plate is located at ρ1L + 3ρ2L/4(ρ1 + v2)

Explanation:

Let area of each half is A and thickness of the plane is t

Xcm = m1x1  + m2x2 / m1 + m2

       = (ρ1 . v/2 )(0) + (ρ2 . v/2)(L/2) / (ρ1 + ρ2) . v/2

       = ρ2 . L/2 / ρ1 + ρ2

       = ρ2 /2 (ρ1 + ρ2) L

Mass of left half = m1 = ρ1  At x1 = L/4

Mass of right half = m2 = ρ2 At x2 = 3L/4

Now apply

Xcm = m1x1  + m2x2 / m1 + m2 = ρ1L+3ρ2L /4(ρ1 + ρ2)

Answer 2

Bravo! Its an IIT Question!

Answer:

Centre of Mass = (d₁ + 3d₂)L/ 4(d₁ + d₂)

Explanation:

Refer to the attached image for the systematic representation of the case.

Given;-

    Density of first half = d₁

    Density of 2nd half = d₂

    The length of the plate = L

Then, breadth of the plate = L

Now;-

The centre of mass of a body is at its centre provided if its mass is uniformly spread. Therefore, for first half, let the centre of mass be at d₁ which is equal to L/4 (r₁) from origin O. Similarly, we consider the centre of mass for second half. Now, the distance of centre of mass for d₂ is equal to 3L/4 (r₂).

Then, we know that;-

∵     Xcm = x₁r₁ + x₂r₂ / m₁ + m₂

Note: Xcm represent the centre of mass in x-axis.

∴     Xcm = m₁ (L/4) + m₂ (3L/4) / m₁ + m₂

       Xcm = d₁(L₂/2)L/4 + d₂(L₂/2)3L/4 / d₁(L₂/2) + d₂(L₂/2)

[since, mass = density × area]

        Xcm = (d₁ + 3d₂) × L/4 / (d₁ + d₂)

        Xcm = (d₁ + 3d₂)L / 4(d₁ + d₂)

Hence, the location of centre of mass of the plate is  (d₁ + 3d₂)L / 4(d₁ + d₂).

Hope it helps! ;-))