The Center of Mass Problems and Solutions

   Problem#1

Four objects are situated along the y axis as follows: a 2.00 kg object is at +3.00 m, a 3.00-kg object is at +2.50 m, a 2.50-kg object is at the origin, and a 4.00-kg object is at –0.500 m. Where is the center of mass of these objects?

Answer:
The x-coordinate of the center of mass is

x_CM = ∑m_ix_i/∑m_i = (0 + 0 +0)/(2.00 kg + 3.00 kg + 4.00 kg)
x_CM = 0

and the y-coordinate of the center of mass is

y_CM = ∑m_iy_i/∑m_i = [(2.00 kg)(3.00 m) + (3.00 kg)(2.50 m) + (2.50 kg)(0)]/(2.00 kg + 3.00 kg + 4.00 kg)

y_CM = 1.00 cm

Problem#2
A water molecule consists of an oxygen atom with two hydrogen atoms bound to it (Fig. 1). The angle between the two bonds is 106°. If the bonds are 0.100 nm long, where is the center of mass of the molecule?

Answer:
Take x-axis starting from the oxygen nucleus and pointing toward the middle of the V.
Then, y_CM = 0 and

x_CM = ∑m_ix_i/∑m_i = [0 + (1.008u)(0.100 nm) cos53.0⁰ + (1.008u)(0.100 nm) cos53.0⁰]/(15.999u + 1.008u + 1.008u)

x_CM = 6,73 pm from the oxygen nucleus

Problem#3
The mass of the Earth is 5.98 x 10²⁴ kg, and the mass of the Moon is 7.36 x 10²² kg. The distance of separation, measured between their centers, is 3.84 x 10⁸ m. Locate the center of mass of the Earth–Moon system as measured from the center of the Earth.

Answer:
Let the x axis start at the Earth’s center and point toward the Moon.

x_CM = ∑m_ix_i/∑m_i = (m₁x₁ + m₂x₂)/(m₁ + m₂)

x_CM = [(5.98 x 10²⁴ kg)(0) + (7.36 x 10²² kg)(3.84 x 10⁸ m)]/(5.98 x 10²⁴ kg + 7.36 x 10²² kg)
x_CM = 4.67 x 10⁶ m from the Earth’s center

Problem#4
A uniform piece of sheet steel is shaped as in Figure 2. Compute the x and y coordinates of the center of mass of the piece.

Answer:
Let A₁ represent the area of the bottom row of squares, A₂ the middle square, and A₃ the top pair.
A₁ = 10.0 cm x 30.0 cm = 300 cm², x₁ = 15.0 cm, y₁ = 5.00 cm
A₂ = 10.0 cm x 10.0 cm = 100 cm², x₂ = 5.00 cm, y₂ = 15.0 cm
A₃ = 10.0 cm x 20.0 cm = 200 cm², x₃ = 10.0 cm, y₃ = 25.0 cm

Then,

x_CM = ∑A_ix_i/∑A_i = (A₁x₁ + A₂x₂ + A₃x₃)/(A₁ + A₂ + A₃)

x_CM = [(300 cm²)(15.0 cm) + (100 cm²)(5.00 cm) + (200 cm²)(10.0 cm)]/(300 cm² + 100 cm² + 200 cm²)

x_CM­ = 11.7 cm

and
y_CM = ∑A_iy_i/∑A_i = (A₁y₁ + A₂y₂ + A₃y₃)/(A₁ + A₂ + A₃)

y_CM = [(300 cm²)(5.00 cm) + (100 cm²)(15.0 cm) + (200 cm²)(25.0 cm)]/(300 cm² + 100 cm² + 200 cm²)

y_CM­ = 13.3 cm    

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