Question

Find the \(x\) coordinate of the centre of mass of five identical masses placed at \(x=2, x=5\), \(x=7, x=9, x=12\)

Short answer

Answer: The x-coordinate of the center of mass of the five identical masses is 7.

Step-by-step solution

List the given x-coordinates

The given \(x\) coordinates are \(x_1=2, x_2=5, x_3=7, x_4=9\), and \(x_5=12\).

Sum the x-coordinates

Calculate the sum of the \(x\) coordinates: \(\sum_{i=1}^5 x_i = x_1 + x_2 + x_3 + x_4 + x_5 = 2 + 5 + 7 + 9 + 12 = 35\)

Calculate the average of the x-coordinates

To find the average of the \(x\) coordinates, divide the sum by the total number of masses: \(X_{cm} = \frac{\sum_{i=1}^5 x_i}{5} = \frac{35}{5} = 7\) The \(x\) coordinate of the center of mass of the five identical masses is \(X_{cm} = 7\).

Key concepts

Physics for Engineers

Physical principles are fundamental in engineering. Conceiving, designing, and analyzing systems hinge on an engineer's ability to understand and apply these principles appropriately.

Center of Mass in Engineering Applications

In the context of physics for engineers, the concept of center of mass is a vital aspect that plays a significant role in various applications such as designing stable structures, understanding the behavior of objects under forces, and in the motion of bodies in aerospace dynamics.

When dealing with structures or vehicles, engineers must ensure that the center of mass leads to a stable configuration to prevent tipping over. In robotics, the center of mass affects how robots balance and move. More advanced applications may involve the center of mass calculations of satellites as they orient themselves in space. Engineering students must grasp this concept to create innovative solutions that are both functional and safe in the physical world.

Mechanics

Mechanics is the branch of physics concerned with the motion of objects and the forces that affect this motion. It is the foundation of classical physics and a core area that engineers must be adept in to analyze the behavior of physical systems.

Within mechanics, statics deals with bodies at rest and in equilibrium -- this is where the center of mass often comes into play. When an object is in static equilibrium, the sum of the forces and moments acting on it equals zero. Identifying the center of mass helps engineers to understand weight distribution and balance of the object.

Center of Mass in Problem Solving

Problem-solving in mechanics often involves calculations like the center of mass to predict the outcome of force applications. The step-by-step exercise provided in the textbook simplifies this concept for engineering students, emphasizing its crucial role in statics and dynamics, subdivisions of mechanics. It is a fundamental skill in engineering education that enables the practical application of theoretical knowledge.

Mass Distribution

Mass distribution refers to how mass is spread out in space within an object or system of objects. It heavily influences the dynamics of motion and the overall behavior of physical systems.

The center of mass is the average location of all the mass in the object, and it is a critical point that behaves as if all the mass were concentrated at that single point for the purpose of analyzing motion.

Important for Safety and Functionality

In engineering, considering mass distribution can be crucial for construction safety, where uneven mass distribution may lead to structural failures, or for vehicles where it affects stability and handling. Understanding mass distribution assists in achieving the desired level of performance while providing essential insights into safety and functionality. The textbook exercise simplifies the calculation for finding the center of mass, by considering identical masses at different positions along a line, making it easier for students to comprehend the more complex situations they'll encounter in their engineering careers.