To find the height of the center of mass of the system consisting of the liquid and the sphere, we need to consider the contributions of both the liquid and the sphere to the overall center of mass. Let's break this down step by step.
Understanding the Components
We have two main components in this problem:
- The Liquid: It has a mass of 5 kg and a density of p.
- The Sphere: It has a mass of 2 kg and a density of 2p.
Position of Each Component
Next, we need to determine the positions of the center of mass for both the liquid and the sphere:
- The center of the sphere is located 3 m from the base of the box.
- The liquid occupies the remaining height of the box, which is 4 m total. Since the liquid fills the box from the base up to the height of the box, we can assume its center of mass is at the midpoint of its height. The height of the liquid is 4 m, so its center of mass is at:
Height of liquid center of mass = 4 m / 2 = 2 m from the base.
Calculating the Center of Mass
The formula for the center of mass (CM) of a system of particles is given by:
CM = (m1 * h1 + m2 * h2) / (m1 + m2)
Where:
- m1: Mass of the liquid = 5 kg
- h1: Height of the center of mass of the liquid = 2 m
- m2: Mass of the sphere = 2 kg
- h2: Height of the center of mass of the sphere = 3 m
Plugging in the Values
Now, substituting the values into the center of mass formula:
CM = (5 kg * 2 m + 2 kg * 3 m) / (5 kg + 2 kg)
CM = (10 kg*m + 6 kg*m) / 7 kg
CM = 16 kg*m / 7 kg
CM = 16/7 m
Final Result
The height of the center of mass of the liquid and sphere system with respect to the base of the box is:
16/7 m
This corresponds to option (3) 16/7 m. This calculation shows how the distribution of mass and the height of each component affect the overall center of mass of the system.
