To determine the location of the center of mass of the system, we need to calculate the weighted average position of all the masses involved.
a. To find the center of mass of the system:
Step 1: Calculate the individual weighted positions for each mass.
Position of mass m1 = 0 (since it is at the left end)
Position of mass m2 = L/2 (since it is at the midpoint of the beam)
Position of mass m3 = L (since it is at the right end)
Position of mass m4 = L (since it is at the right end)
Step 2: Calculate the total mass of the system.
Total mass (m_total) = m1 + m2 + m3 + m4
Step 3: Calculate the weighted average position.
Center of mass position (x_cm) = (m1 * 0 + m2 * (L/2) + m3 * L + m4 * L) / m_total
Substituting the given values:
x_cm = (57 * 0 + 92 * (2.7/2) + 67 * 2.7 + 12 * 2.7) / (57 + 92 + 67 + 12)
Simplifying the equation gives us the answer for part (a).
Once you have calculated the center of mass, you can proceed to part (b), (c), and (d) using the new conditions stated in the problem.