Final answer:
To find the center of mass of the rod, we need to consider the mass and position of each segment of the rod. The total distance from the left end to the center of mass is calculated by summing the products of the masses and their respective distances from the left end, divided by the total mass of the rod.
Step-by-step explanation:
To find the center of mass of the rod, we need to consider the mass and position of each segment of the rod. The total distance from the left end to the center of mass is given by the sum of the products of the masses and their respective distances from the left end, divided by the total mass of the rod.
For the iron segment, which is 32 cm long and has a density of 8 g/cm³, the mass can be calculated by multiplying the density by the volume (length × cross-sectional area). The volume is (32 cm × 1 cm² = 32 cm³).
Similarly, for the aluminum segment, which is 41 cm long and has a density of 2.7 g/cm³, the mass can be calculated by multiplying the density by the volume (41 cm × 1 cm² = 41 cm³).
After calculating the masses of both segments, we can find the center of mass by dividing the sum of the products of mass and distance by the total mass of the rod.
Let's calculate it:
Mass of iron segment = density × volume = 8 g/cm³ × 32 cm³ = 256 g
Mass of aluminum segment = density × volume = 2.7 g/cm³ × 41 cm³ = 110.7 g
Total mass of the rod = mass of iron segment + mass of aluminum segment = 256 g + 110.7 g = 366.7 g
Center of mass = (mass of iron segment × distance of iron segment from left end + mass of aluminum segment × distance of aluminum segment from left end) / total mass of the rod = (256 g × 16 cm + 110.7 g × 73 cm) / 366.7 g = 116.38 cm
Therefore, the center of mass of the rod is 116.38 cm from the left end.