Questions: At time t=0, a system consists of three spheres with different masses located at the distances shown on the figure. a. Draw a free body diagram for the center sphere, indicating the gravitational force acting on it due to the spheres surrounding it. Each vector should consist of an arrow starting on and pointing away from the center sphere. The length of the vectors should be drawn relative to their magnitudes. Make sure to label the arrows.

Solution Steps

The center sphere (2 kg) experiences gravitational forces from the other two spheres (10 kg and 5 kg). The force due to the 10 kg sphere acts to the left (negative x-direction), and the force due to the 5 kg sphere acts to the right (positive x-direction). The center sphere also experiences a gravitational force upwards (positive y-direction) and a gravitational force downwards (negative y-direction). Here the picture doesn't specify any other masses, so we only consider the two masses given in the horizontal plane.

The magnitude of the gravitational force between two objects is given by \(F = G\frac{m_1 m_2}{r^2}\), where G is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the objects, and \(r\) is the distance between their centers.

Force due to the 10 kg sphere: \(F_{10} = G\frac{(2)(10)}{50^2} = \frac{20G}{2500} = \frac{2G}{250}\) Force due to the 5 kg sphere: \(F_5 = G\frac{(2)(5)}{20^2} = \frac{10G}{400} = \frac{G}{40}\)

Since \(\frac{G}{40} > \frac{2G}{250}\), the force due to the 5 kg sphere is larger than the force due to the 10 kg sphere.

The free body diagram for the center sphere (2 kg) will have two arrows:

1. An arrow pointing to the left representing the force due to the 10 kg sphere (\(F_{10}\)). This arrow should be shorter.

2. An arrow pointing to the right representing the force due to the 5 kg sphere (\(F_5\)). This arrow should be longer.

   <----F₁₀ 2 kg F₅----> 10 kg-------------------●-------------------5 kg 50 m 20 m

Final Answer

  <----F₁₀     2 kg       F₅---->