Q: A uniform beam of mass 5.0 kg holds up two masses, 22 kg and 15 kg, placed at distances 1.0 m and 4.0 m from the left edge, respectively. What is the center of gravity of the system from the left edge?

$$x_{CG} = \frac{(22 \times 1.0) + (5.0 \times 3.0) + (15 \times 4.0)}{22 + 5.0 + 15}$$

Q: Given three blocks with masses 4 kg, 6 kg, and 8 kg located at positions (2, 0), (0, 1), and (2, 1) respectively, what is the y-coordinate of the center of mass?

$$y_{CM} = \frac{(4 \times 0) + (6 \times 1) + (8 \times 1)}{4 + 6 + 8}$$

Question 1

Which statement best describes the center of mass of a homogeneous symmetrical object?

Accepted Answer

It is the geometric midpoint of the object, often at the intersection of symmetry axes.

Question 2

For a system of particles in two dimensions, which formula correctly gives the x-coordinate of the center of mass?

Accepted Answer

$$x_{CM} = \frac{\sum m_i x_i}{M_{Total}}$$

Question 3

Why does the Leaning Tower of Pisa not topple over, despite its tilt?

Accepted Answer

Because the vertical line from its center of gravity passes through its base.

Question 4

Which of the following is true about the center of gravity (CG) of an object?

Accepted Answer

The CG can be located at a point where no actual material exists.

Question 5

A uniform beam of mass 5.0 kg holds up two masses, 22 kg and 15 kg, placed at distances 1.0 m and 4.0 m from the left edge, respectively. What is the center of gravity of the system from the left edge?

Accepted Answer

$$x_{CG} = \frac{(22 	imes 1.0) + (5.0 	imes 3.0) + (15 	imes 4.0)}{22 + 5.0 + 15}$$

Question 6

Which condition must be met for an object to remain stable and not topple over?

Accepted Answer

The vertical line from the center of gravity must pass through the base of the object.

Question 7

A system consists of four particles, each of mass 2 kg. Two are located at 3 cm and 5 cm on the +x-axis, and two at 2 cm and 4 cm on the -x-axis. What is the x-coordinate of the center of mass?

Accepted Answer

$$x_{CM} = 0.5\ \mathrm{cm}$$

Question 8

Which of the following best explains why a lifebuoy's center of gravity may be located where there is no material?

Accepted Answer

Because the center of gravity depends on the distribution of mass, not the presence of material.

Question 9

Given three blocks with masses 4 kg, 6 kg, and 8 kg located at positions (2, 0), (0, 1), and (2, 1) respectively, what is the y-coordinate of the center of mass?

Accepted Answer

$$y_{CM} = \frac{(4 	imes 0) + (6 	imes 1) + (8 	imes 1)}{4 + 6 + 8}$$

Question 10

Which of the following scenarios describes a neutral equilibrium for an object?

Accepted Answer

When the center of gravity is above the base and the object neither returns nor continues to topple after being tilted.

Question 11

If a vertical line drawn from the center of gravity of an object does NOT pass through its base, what will happen?

Accepted Answer

The object will topple over.

Question 12

Which of the following is the correct formula for the y-coordinate of the center of mass for a system of particles?

Accepted Answer

$$y_{CM} = \frac{\sum m_i y_i}{M_{Total}}$$

Question 13

A system consists of two masses, $$m_1$$ and $$m_2$$, separated by a distance $$d. If$$ $$m_1$$ \u003e $$m_2$$, where is the center of mass located?

Accepted Answer

Closer to $$m_1$$

Question 14

Which of the following objects will have its center of mass at the intersection of its symmetry axes?

Accepted Answer

A homogeneous triangle

Question 15

A car is parked on a slope. Which factor most affects whether it will topple or remain stable?

Accepted Answer

Whether the vertical line from the center of gravity passes through the base of support

Question 16

Which of the following is a real-world example where the center of mass is outside the physical material of the object?

Accepted Answer

A hollow ring

Question 17

If the mass of each block in a system is doubled, but their positions remain the same, what happens to the center of mass?

Accepted Answer

It remains unchanged.

Question 18

Which of the following best describes the difference between center of mass and center of gravity near Earth's surface?

Accepted Answer

They coincide because the gravitational field is nearly uniform.

Question 19

A gymnast balances on a beam. To avoid toppling, what must be true about her center of gravity?

Accepted Answer

It must be directly above the beam.

Center of Mass and Center of Gravity: Concepts, Stability, and Calculations

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