Question

Masses of \(3 \mathrm{~kg}, 2 \mathrm{~kg}, 2 \mathrm{~kg}\) and \(4 \mathrm{~kg}\) are located at points with coordinates \((1,3),(-2,0)\), \((4,-1)\) and \((-3,4)\), respectively. Calculate the coordinates of the centre of mass.

Short answer

Answer: The coordinates of the center of mass are (-5/11, 23/11).

Step-by-step solution

1. Find the total mass of the system

We will add up the masses given to get the total mass of the system. $$ M = m_1 + m_2 + m_3 + m_4 = 3kg + 2kg + 2kg + 4kg = 11kg $$

2. Find the weighted coordinates

Now we will multiply the coordinates of each mass by their respective masses. $$ \boldsymbol{r}_1 m_1 = (1,3)\cdot 3kg = (3,9) \\ \boldsymbol{r}_2 m_2 = (-2,0)\cdot 2kg = (-4,0) \\ \boldsymbol{r}_3 m_3 = (4,-1)\cdot 2kg = (8,-2) \\ \boldsymbol{r}_4 m_4 = (-3,4)\cdot 4kg = (-12,16) $$

3. Find the sum of the weighted coordinates

Now we add up the products from the previous step. $$ \boldsymbol{R}_{sum} = \boldsymbol{r}_1 m_1 + \boldsymbol{r}_2 m_2 + \boldsymbol{r}_3 m_3 + \boldsymbol{r}_4 m_4 = (3,9) + (-4,0) + (8,-2) + (-12,16) = (-5,23) $$

4. Calculate the coordinates of the center of mass

Divide the sum of weighted coordinates by the total mass of the system to find the coordinates of the center of mass. $$ \boldsymbol{R}_{cm} = \frac{\boldsymbol{R}_{sum}}{M} = \frac{(-5,23)}{11kg} = \left(-\frac{5}{11}, \frac{23}{11}\right) $$ The coordinates of the center of mass are \(\boldsymbol{R}_{cm} = \left(-\frac{5}{11}, \frac{23}{11}\right)\).

Key concepts

Centre of Mass

In mechanics, the centre of mass (also referred to as the center of gravity) is a crucial concept that represents the average location of all the mass in a system. Imagine it as a balancing point—a hypothetical point where the entire mass of an object or system could be concentrated without altering its motion under gravity.

To find the centre of mass of a system of particles, we calculate it as the mass-weighted average of their positions. The calculation involves multiplying the mass of each particle by its position (its coordinates), summing all those products together, and then dividing by the total mass of the system. This gives us the coordinates of the centre of mass.

The centre of mass is particularly useful because it simplifies complex systems into a single point which can be analyzed to predict the motion of the entire system under various forces.

Mechanics

Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements. It encompasses the study of motion, forces, energy, and the interactions between objects.

Understanding centre of mass is fundamental in mechanics because it directly relates to the motion of an object. For instance, when you throw a javelin, its trajectory is largely determined by the motion of its centre of mass. Mechanics uses the principles of conservation of momentum and energy to analyze and predict such movements. When governing the motion of extended bodies, rather than treating them as a collection of individual particles, mechanics simplifies the problem by focusing on the motion of the centre of mass.

Coordinate System

A coordinate system in physics provides a method to identify the position of a point or a particle in space with numerical coordinates. The most common system is the Cartesian coordinate system, which specifies the position of a point by distances from perpendicular axes, usually labeled as x and y for two-dimensional space, and including z for three-dimensional space.

When solving for the centre of mass, we utilize a coordinate system to describe the positions of all the masses involved. This allows us to apply mathematical methods to analyze the spatial distribution of mass. Knowing the coordinates of individual masses and their corresponding weights (mass values), we can calculate the coordinates of the centre of mass effectively.

Mass Distribution

Mass distribution refers to how mass is spread out in space. The distribution of mass affects the balance, stability, and motion of objects. In our exercise, we considered point masses with specific coordinates and values, representing a discrete mass distribution. This is a simplification often used in physics problems.

Determining the centre of mass in a system with a given mass distribution involves considering the locations and magnitude of all the masses. Unequal distribution of mass leads to the centre of mass being closer to the larger mass. This concept is essential in various fields, including engineering, where it is used to ensure the stability of structures by designing around the centre of mass.