Question

In the letter E whose dimensions are given in the figure, origin is at the bottom left corner. Area of each rectangle is 12 cm2 and area of the square is 4 cm2. Assume that the weights are proportional to areas. Then, centre of gravity of the E frame will be at:

- (1 cm,3 cm)
- (2 cm,2 cm)
- (2.4 cm,5 cm)
- (3 cm,6 cm)

Solution

The correct option is **C** (2.4 cm,5 cm)

Divide the whole E frame into three rectangles and one square as shown in the figure below, with origin at O.

Given that weights are proportional to area, hence centre of gravity for the body will coincide with COM of the body.

Let G1,G2,G3 and G4 be the centres of mass of the four parts and A1,A2,A3 and A4 be the areas of the rectangles and square shapes respectively.

Then,

G1⇒(x1,y1)=(3,9)

G2⇒(x2,y2)=(3,1)

G3⇒(x3,y3)=(1,5)

G4⇒(x4,y4)=(3,5)

and

A1=12 cm2, A2=12 cm2

A3=12 cm2, A4=4 cm2

Applying the formula for xCM and yCM of E frame:

xCM=A1x1+A2x2+A3x3+A4x4A1+A2+A3+A4=(12×3)+(12×3)+(12×1)+(4×3)12+12+12+4

∴xCM=9640 cm=2.4 cm

Simillarly;

yCM=A1y1+A2y2+A3y3+A4y4A1+A2+A3+A4=(12×9)+(12×1)+(12×5)+(4×5)12+12+12+4

∴yCM=20040 cm=5 cm

Hence coordinates of centre of gravity of E frame (x,y)=(2.4 cm,5 cm)

Divide the whole E frame into three rectangles and one square as shown in the figure below, with origin at O.

Given that weights are proportional to area, hence centre of gravity for the body will coincide with COM of the body.

Let G1,G2,G3 and G4 be the centres of mass of the four parts and A1,A2,A3 and A4 be the areas of the rectangles and square shapes respectively.

Then,

G1⇒(x1,y1)=(3,9)

G2⇒(x2,y2)=(3,1)

G3⇒(x3,y3)=(1,5)

G4⇒(x4,y4)=(3,5)

and

A1=12 cm2, A2=12 cm2

A3=12 cm2, A4=4 cm2

Applying the formula for xCM and yCM of E frame:

xCM=A1x1+A2x2+A3x3+A4x4A1+A2+A3+A4=(12×3)+(12×3)+(12×1)+(4×3)12+12+12+4

∴xCM=9640 cm=2.4 cm

Simillarly;

yCM=A1y1+A2y2+A3y3+A4y4A1+A2+A3+A4=(12×9)+(12×1)+(12×5)+(4×5)12+12+12+4

∴yCM=20040 cm=5 cm

Hence coordinates of centre of gravity of E frame (x,y)=(2.4 cm,5 cm)

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