Chapter 13 Problem 006 In The Figure A Square Of Edge Length

Chapter 13 Problem 006in The Figure A Square Of Edge Length 240 Cm

In the figure, a square of edge length 24.0 cm is formed by four spheres of masses m₁ = 5.30 g, m₂ = 2.50 g, m₃ = 0.500 g, and m₄ = 5.30 g. Determine the net gravitational force exerted on a central sphere with mass m₅ = 2.10 g by these four spheres, expressed in unit-vector notation.

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The problem involves calculating the net gravitational force on a central sphere due to four surrounding spheres positioned at the corners of a square. To approach this problem, we employ Newton's law of universal gravitation, which states that the force between two masses is proportional to their product and inversely proportional to the square of the distance between them. The problem provides the edge length of the square and the masses of four spheres at the corners, with a central sphere at the square’s center.

First, it is essential to determine the positions of the four spheres relative to the central sphere. Since they form a square with an edge length of 24.0 cm, and the central sphere is at the center, each sphere is located at a distance of half the diagonal of the square from the center. The diagonal \(d\) of the square is given by:

d = \sqrt{2} \times 24.0\, \text{cm} ≈ 33.94\, \text{cm}

Half the diagonal, which is the distance from the center to each corner, is therefore:

r = d/2 ≈ 16.97\, \text{cm} = 0.1697\, \text{m}

Next, we determine the direction of forces exerted by each sphere on the central sphere. Since the four corner spheres are symmetrically placed, their positions relative to the center can be represented in a coordinate system, with each sphere at positions corresponding to the vertices of a square centered at the origin.

The positions of the corner spheres (assuming the center at (0,0)) are:

  • Sphere 1 (top-right corner): ( +r, +r )
  • Sphere 2 (top-left corner): ( -r, +r )
  • Sphere 3 (bottom-left corner): ( -r, -r )
  • Sphere 4 (bottom-right corner): ( +r, -r )

Calculating the force exerted by each sphere involves the following steps:

  1. Calculate the magnitude of the gravitational force between each sphere and the central sphere:
F = G  m_center  m_i / r^2

Where:

  • G is the gravitational constant (\(6.674 \times 10^{-11}\, \text{Nm}^2/\text{kg}^2\))
  • m_center = 2.10 g = 0.00210 kg
  • m_i are the masses of the corner spheres.
m₁ = 5.30\, \text{g} = 0.00530\, \text{kg} \\

m₂ = 2.50\, \text{g} = 0.00250\, \text{kg} \\


m₃ = 0.500\, \text{g} = 0.000500\, \text{kg} \\


m₄ = 5.30\, \text{g} = 0.00530\, \text{kg}

The force on the central sphere due to each corner sphere can be expressed in vector form as:

F_i = F_i * r̂_i

Where r̂_i is the unit vector pointing from the corner sphere to the central sphere. For each sphere at position (x_i, y_i), the unit vector is:

r̂_i = (x_i, y_i) / r

Due to symmetry, the forces along the axes will cancel out in pairs, and the net force will be the vector sum of the individual forces. Explicitly, the force calculations will be performed for each sphere, taking into account their positions and masses.

Finally, summing all force vectors yields the net gravitational force on the central sphere in unit-vector notation. The calculations involve substituting the known values into Newton’s law, computing the force magnitudes, and applying the vector components accordingly.

References

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