Question

find the centre of mass of metalic Letter E of uniform density whose dimensions are given in the figure take the origin at the bottom of the left corner . the width of the letter is 4 cm everywhere​

Answer 1

Answer:

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Answer 2

Given:

Letter E with dimensions 12cm, 20cm, and 4 cm and uniform density of the material

To Find:

The center of mass of the letter

Solution:

Since the thickness and density are the same,

Center of mass of a system = ∑A.r/∑A

Here A = area of the parts and r is the position vector of the parts

We can divide the letter E into 4 parts, 3 the horizontal bars and 1 is vertical bar.

Area 1 = 12 X 4 = 48 cm²

Area 2 = (20-4) X 4 = 16 cm²

Area 3 = (20 - 16) X 4 = 16 cm²

Area 4 = 12 X 4 = 48 cm²

Taking the edge of the letter as the origin (0,0) and using geometry, the COM of parts 1, 2, 3, and 4 are (6,2), (2,10), (6,12), and (7,19) respectively.

X coordinate of Center of mass = ∑Ax/∑A

= A₁X₁ + A₂X₂ + A₃X₃ + A₄X₄ / A₁ + A₂ + A₃ + A₄

= 48 X 6 + 16 X 2 + 16 X 6 + 48 X 7 / 48 + 16 + 16 + 48

= 120 + 128 + 96 + 336 / 176

= 680 / 176

= 3.9 (Approximated)

Similarly, the Y coordinate = ∑Ay/∑A

= A₁Y₁ + A₂Y₂ + A₃Y₃ + A₄Y₄ / A₁ + A₂ + A₃ + A₄

= 48 X 2 + 16 X 10 + 16 X 12 + 48 X 19 / 48 + 16 + 16 + 48

= 96 + 640 + 192 + 912 / 176

= 1840 / 176

= 10.5(Approximated)

Hence, the coordinates of the center of mass are (3.9,10.5).