Center of Mass & Simple Balance: Videos & Practice Problems

The center of mass (CM) of an object determines its stability on a surface. For objects with uniform mass distribution, the CM is at the geometric center. An object will tilt if its CM extends beyond the support edge. The CM can be calculated using the equation: $\frac{\sum m_i{x}_i}{\sum m}$. In non-uniform mass distributions, the CM shifts towards the heavier side, affecting balance and stability.

Center of Mass & Simple Balance

The relationship between an object's center of mass and its stability on a surface is crucial in understanding how objects balance. The weight of an object, represented as \( mg \), acts at its center of gravity, which for most practical purposes can be considered the same as the center of mass. This is particularly true for objects with uniform mass distribution, where the mass is evenly spread throughout the object. In such cases, the center of mass is located at the geometric center of the object.

When analyzing whether an object will tilt or remain balanced, it is essential to consider the position of the center of mass relative to the support point. If the center of mass is within the boundaries of the support, the object will remain in static equilibrium and not tilt. Conversely, if the center of mass extends beyond the edge of the support, the object will begin to tilt, indicating that it is no longer in equilibrium.

To determine the center of mass for a system of objects, the formula used is:

\( x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \)

where \( m_i \) represents the mass of each object and \( x_i \) is the position of each object along the x-axis. For two objects, this can be expressed as:

\( x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \)

In practical scenarios, such as a plank supported at two points with a person walking on it, the position of the center of mass changes as the person moves. For example, if a 20 kg plank is 10 meters long and supported at one end and 3 meters from the other end, the center of mass of the system (plank plus person) must remain within the support points to prevent tipping. If the person has a mass of 60 kg, the goal is to find the maximum distance the person can walk to the right of the rightmost support without causing the plank to tip.

By setting the center of mass of the system equal to the position of the rightmost support, we can solve for the maximum distance the person can walk. This involves substituting the known values into the center of mass equation and solving for the unknown distance. The calculations reveal that the person can only walk approximately 0.67 meters beyond the rightmost support before the plank tips, illustrating the significant impact of mass distribution on stability.

Understanding these principles is essential for solving problems related to static equilibrium and the behavior of objects under various conditions. By applying the center of mass concept, one can predict and analyze the stability of different systems effectively.

Non-Uniform Mass Distributions (Find Center of Mass)

In the study of rigid bodies, understanding mass distribution is crucial. Typically, we assume a uniform mass distribution, meaning the mass is evenly spread throughout the object, allowing us to locate the center of mass at the geometric center. However, many real-world objects, such as the human body, exhibit non-uniform mass distribution, where mass is not evenly distributed. This can significantly affect the location of the center of mass and the forces acting on the object.

For instance, consider a problem involving an 80-kilogram man who is 2 meters tall. In this scenario, we recognize that the human body has a non-uniform mass distribution, with more mass concentrated towards the upper body due to the arms and torso. To find the center of mass, we can analyze the forces acting on the man when he stands on a board supported by two scales. The left scale measures a normal force of 320 N, while the right scale measures 480 N. This difference indicates that the center of mass is closer to the man's head, as the right scale, which is under the heavier side, exerts more force.

To determine the exact position of the center of mass, we can set up an equilibrium problem. The sum of the vertical forces must equal zero, which confirms that the upward forces from the scales balance the downward gravitational force (weight) of the man. However, this alone does not provide information about the center of mass's location.

Next, we analyze the torques acting on the system. By selecting a pivot point (typically where one of the forces acts), we can simplify our calculations. The torque due to the weight of the man (mg) will create a clockwise rotation, while the torque from the right normal force will create a counterclockwise rotation. The equation for torque is given by:

$$\tau = F \cdot r \cdot \sin(\theta)$$

In this case, since both forces act perpendicular to their respective distances, the sine of the angle is 1, simplifying our calculations. The distance from the pivot to the center of mass is represented as \(2 - x\), where \(x\) is the distance from the man's head to the center of mass. The distance from the pivot to the right normal force is simply 2 meters.

Setting up the torque equation, we have:

$$480 \cdot 2 = 80 \cdot 10 \cdot (2 - x)$$

Solving this equation leads to:

$$960 = 800 - 800x$$

Rearranging gives:

$$800x = 640$$

Thus, we find:

$$x = 0.8 \text{ meters}$$

This result indicates that the center of mass is located 0.8 meters from the man's head, confirming our expectation that it is closer to the head than the feet. The total height of the man is 2 meters, so the distance from the feet to the center of mass is 1.2 meters. This analysis illustrates the importance of understanding mass distribution in determining the center of mass and the resulting forces acting on an object.

A 70 kg, 1.90 m man doing push-ups holds himself in place making 20° with the floor. His feet and arms are, respectively, 1.15 m below and 0.4 m above his center of mass. You may model him as a thin, long board, and assume his arms and feet are perpendicular to the floor. How much force does the floor apply to each of his hands? (Use g=10 m/s².) 

• A. 260 N
• B. 472 N
• C. 519 N
• D. 943 N

Answer: (A)

Here’s what students ask on this topic:

What is the center of mass and how is it calculated?

The center of mass (CM) is the point in an object where its mass is considered to be concentrated. For objects with uniform mass distribution, the CM is at the geometric center. The CM can be calculated using the equation:

$\frac{\sum m_i{x}_i}{\sum m_i}$

where $m_i$ is the mass of each object and $x_i$ is the position of each object. This formula helps in finding the weighted average position of the mass in a system.

How does the center of mass affect an object's balance?

The center of mass (CM) significantly affects an object's balance. If the CM is within the base of support, the object will remain stable and balanced. However, if the CM extends beyond the support edge, the object will tilt and become unstable. This principle is crucial in designing stable structures and understanding the balance of various objects.

What is the difference between center of mass and center of gravity?

The center of mass (CM) is the point where the mass of an object is concentrated, while the center of gravity (CG) is the point where the gravitational force acts. In most cases, especially near the Earth's surface, CM and CG are considered the same. However, in non-uniform gravitational fields, they can differ. For most physics problems, treating them as the same simplifies calculations.

How do you find the center of mass for a system of particles?

To find the center of mass (CM) for a system of particles, use the equation:

$\frac{\sum m_i{x}_i}{\sum m_i}$

where $m_i$ is the mass of each particle and $x_i$ is the position of each particle. This formula calculates the weighted average position of the mass in the system, giving the CM's location.

What happens to the center of mass in non-uniform mass distributions?

In non-uniform mass distributions, the center of mass (CM) shifts towards the heavier side of the object. This affects the object's balance and stability. For example, in a human body, the CM is not at the geometric center but is skewed towards the torso due to the uneven mass distribution. This shift must be considered in calculations involving balance and stability.