# Worksheet: Center of Mass

In this worksheet, we will practice calculating the location of the center of mass of a system of objects with various masses and positions relative to each other.

**Q1: **

Three point masses are placed at the corners of a triangle as shown in the accompanying figure. The origin of the system is defined to be at the position of the 150 g mass, with displacement to the right of the origin corresponding to positive values and displacement above the origin corresponding to positive values.

Find the coordinate of the center of mass system.

Find the coordinate of the center of mass system.

**Q2: **

If half of the population of Earth were to somehow be transferred to the Moon, the position of the center of mass of the Earth-Moon system would be slightly changed. Assume that the average mass of a human is 60.0 kg and the human population is people. Use kg for the mass of Earth, kg for the mass of the Moon, and m as the radius of the Moon’s orbit. Assume that the human population is evenly distributed over either Earth’s surface or the Moon’s surface. What is the magnitude of the change of the center of mass of the Earth-Moon system?

**Q3: **

The structure shown has a uniform thickness of 20 cm, and a uniform density of 1.0 g/cm^{3}. Assume an origin at the floor and at the structure’s centerline.

What is the horizontal distance from the origin of the center of mass of the object?

What is the vertical distance from the origin of the center of mass of the object?

**Q6: **

Two particles of masses 2.0 kg and 4.0 kg move in uniform circles with radii of 5.0 cm and cm respectively. The -coordinate of the particle moving in the 5.0 cm radius circle is given by and the -coordinate is given by . The -coordinate of the center of mass of the particles is given by and the -coordinate of the center of mass of the particles is given by . Find .

**Q8: **

A system comprised of a sphere and a cylinder can be arranged in different ways, as shown in the diagram. The cylinder has a length cm and a radius cm. The sphere has a radius cm. The cylinder and the sphere have the same density. In arrangement , the axis of the cylinder along its length passes through the center of the sphere. In arrangement , the axis of the cylinder along the vertically directed radius of its circular face, horizontally half-way along the cylinder’s length, passes through the center of the sphere.