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11 days ago

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Three beads are placed on the vertices of an equilateral triangle of side d = 3.4cm. The first bead of mass m1=140gis placed on

the top vertex. The second bead of mass m2=45g is placed on the left vertex. The third bead of mass m3=85g is placed on the right vertex.
(a) Write a symbolic equation for the horizontal component of the center of mass relative to the left vertex of the triangle.

(b) Find the horizontal component of the center of mass relative to the left vertex, in centimeters.

(c) Write a symbolic equation for the vertical component of the center of mass relative to the base of the triangle.

(d) Find the vertical component of the center of mass relative to the base of the triangle, in centimeters.

1 answer:

serg [3.2K]11 days ago

8 0

Answer:

Xcm = 1.95 cm and Ycm = 1.76 cm

Explanation:

The mass center concept is quite significant.

R cm = 1/M ∑ $m_{i}$ $r_{i}$

Here, ri and mi are the positions of the masses from a chosen reference point, while M represents the total mass.

First, we will calculate the total mass.

M = m₁ + m₂ + m₃

M = 140 + 45 + 85

M = 270 g

Now, let’s determine the position for each vertex.

For Point 1, the top vertex has a triangle side length d.

R₁ = d / 2 i ^ + d j ^

R₁ = (1.7 cm i ^ + 3.4 j ^) cm

For Point 2, the left vertex. What is the origin of our reference?

R₂ = 0

For Point 3, the right vertex.

R₃ = d i ^

R₃ = 3.4 i ^ cm

a) The x component of the center of mass is calculated as follows:

Xcm = 1 / M (m₁ x₁ + m₂ x₂ + m₃ x₃)

Xcm = 1 / M (m₁ d / 2 + 0 + m₃ d)

Xcm = d / M (m₁ / 2 + m₃)

b) Now we calculate the x center of mass component:

Xcm = 1/270 (1.7 140 + 0 + 3.4 85)

Xcm = 238/270

Xcm = 1.95 cm

c) We will find the vertical center of mass component.

Ycm = 1 / M (m₁ y₁ + m₂ y₂ + m₃ y₃)

Ycm = 1 / M (m₁ d + 0 + 0)

Ycm = m₁ / M d

d) Now we will compute:

Y cm = 1/270 (140 3.4 + 0 + 0)

Ycm = 1.76 cm

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