In this explainer, we will learn how to find the position of the center of mass of a set of particles arranged in a two-dimensional plane.
Often in mechanics, we consider the motion of an object as though that object were a point mass—an object that has mass, but no length, width, or height. Even when we solve problems involving real-world objects such as tennis balls or boxes, we often model them as point masses. Doing this makes solving a lot of problems simpler.
However, real-world objects do have length, width, and height, and the shape of an object can affect its motion. But, depending on the scenario, we can often still apply many of the formulae and methods we use for point masses to more complex objects. We can do this because, for many problems, we can model a complex object as though it were a point mass at what is called its center of mass, which is also sometimes called its center of gravity.
The center of mass of a set of particles can be thought of as the average of the positions of all of the particles weighted by their masses. So, for example, imagine two particles with position vectors and and masses and , as shown in the diagram of a two-dimensional system.
The position vector of the center of mass of the two particles, , is given by
Notice that this is different from the geometric center of the particles. The geometric center of the particles would just be equal to . The center of mass would only be in the same position as the geometric center in the special case where all of the particles have the same mass.
The center of mass is weighted by the masses of the particles. This means that if the mass of the first particle, , were less than that of the second particle, the center of mass would be closer to the second particle than the first, which is the case in the diagram.
If a third particle were added to the system, with a position vector and a mass , we would have to add a third term to the numerator of the fraction that is the product of the mass and the position vector. The denominator of the fraction is just the total mass of all of the particles, so this formula would become
If we call the total mass , we can write this as
We can see a pattern forming here. As we add more particles to the system, we keep adding terms of the mass of each particle times its position vector to the bracketed expression.
Definition: Center of Mass of a System of Particles
For a system of particles, where the particle in the system has position vector and mass , the position vector of the center of mass of the system, , is given by where is the total mass of all of the particles.
Let us now consider the center of mass of a rigid body about the origin point of a coordinate system. Suppose we are given the weights of the particles forming the rigid body, which are , situated at the position vectors , respectively, relative to the origin. We can find the position of the center of mass, , relative to the origin using the formula which can also be rewritten using summation notation as
We can see how this formula links back to the original formula involving just the masses. We know that we can write each of the weights as the product of their mass and the gravitational constant, :
If we were to substitute each of these into the formula we have just found for finding the position of the center of mass, then each of the terms would be a multiple of . If we were to cancel this factor, then we would end up with which is the same as the original formula we found. We can now consider - and -components of the position of the center of mass. We can let be the horizontal component of and be the vertical component of . Then, if are the horizontal components and are the vertical components of , respectively, we can form the following two formulas by equating the horizontal and vertical components of our vector for the center of mass:
This formula would also work if we were given the weights of the particles instead of the mass, since if we multiply each term in both of the fractions by , then can rewrite each as .
If you are presented with a problem where you are given either the coordinates, or the position vectors for a system of particles, and you are asked to find the center of mass of the system, you can solve this directly using the formula, or it can be helpful to present the information in a table to help you form the equations that you need to solve.
For example, consider particles and whose masses are 3 kg and 4 kg, respectively, and whose position vectors are and respectively. We can present this information in a table as follows.
| Mass | 3 | 4 | 7 |
|---|---|---|---|
| -coordinates | 3 | 2 | |
| -coordinates | 4 | 5 |
Here, 7 is the total mass of the system and represents the position vector of the center of mass of the system. From the table, we can then form two equations allowing us to calculate the coordinates for the center of mass:
Solving these equations gets us to a position vector of for the center of mass of the system. In this explainer, we will predominantly solve the examples directly using the formula.
We will now look at an example of how we can use these formulas to find the position of the center of mass of a system in two dimensions.
Example 1: Finding the Center of Mass of a Two-Dimensional System
In the given figure, three weights of magnitudes 2 N, 5 N, and 3 N are placed on the vertices of an equilateral triangle of side length 8 cm.
Find the center of mass of the system.
Answer
We can start by putting the data for the coordinates of each of the weights in a table.
| Weight | 2 N | 5 N | 3 N |
|---|---|---|---|
| -Coordinate | 0 | 0 | |
| -Coordinate | 0 | 8 | 4 |
In order to find the -coordinate of the center of mass, we can use the formula
Substituting our values into this formula, we have
We can now consider the -coordinate, using a similar formula:
Substituting from the table, we have
Hence the position of the center of mass of the system is .
Now, let us look at another example.
Example 2: Determining the Position Vector of the Center of Mass of Three Discrete Masses
Suppose three masses of 1 kg, 4 kg, and 6 kg are located at points whose position vectors are , , and . Determine the position vector of the center of mass for this system of masses.
Answer
We can use the formula to find the position vector for the center of mass of the system, , where is the total mass of the system, is the mass of object , and is the position vector of object . is the total mass, so
Now, let us substitute the values for the masses and position vectors of the objects into the above formula:
Now, we just need to simplify the expression:
The position vector of the center of mass is .
With questions involving finding the position vector, or coordinates of the the center of mass of a system, it is often worth considering the geometry of the system prior to finding the solution. For example, in a two-dimensional system, if the system of particles all lie on a horizontal or vertical line, then we know that the center of mass must also lie on this line, and, therefore, we only have one unknown coordinate.
This is demonstrated in our next example.
Example 3: Finding the Position of the Center of Mass for a System of Particles
Four particles of masses 9 kg, 10 kg, 4 kg, and 7 kg are placed on the -axis at the points , , , and respectively. What is the position of the center of mass of the four particles?
Answer
Here, we can start by representing the coordinates of each of the particles as vectors and then solving the problem directly using the formula, or we can simplify the problem by first considering its geometry. For completeness, let us demonstrate both methods here.
Method 1
First, we convert the coordinates to vectors: , , , and .
We can use the formula to find the center of mass of the system, , where is the total mass of the system, is the mass of object , and is the position vector of object . is the total mass, so
Now, let us substitute the values for the masses and position vectors of the objects into the above formula:
Now, we just need to simplify the expression:
The position vector of the center of mass is , which is when written as a set of coordinates.
Method 2
Notice first that all of the points lie on the -axis, and, therefore, the center of mass of the system must also lie on the -axis and must have coordinates .
Second, we need to calculate the total mass of the system:
Now, we can present the information in a table, as follows.
| Mass | 9 | 10 | 4 | 7 | 30 |
|---|---|---|---|---|---|
| -coordinates | 4 | 3 | 8 | 1 |
From the table, we can form the following equation:
Simplifying gives us which we can solve as follows:
Therefore, the coordinates of the center of mass are .
Sometimes, we will be asked to calculate an unknown mass in a system given that we know its center of mass. We will demonstrate this in our next example.
Example 4: Finding Unknown Discrete Masses given the Coordinates of Their Center of Mass
The points , , and on the -axis are occupied by three solids of masses 9 kg, 6 kg, and kg respectively. Determine the value of given the center of mass of the system is at the point .
Answer
There are two methods we could use to solve this problem. The first is to form and complete a table of the masses in the system and the second is to convert the positions of the masses to vector form and solve using the formula. We will be showing both methods here.
Method 1
We can start by forming a table containing the mass, -coordinate, and -coordinate of each of the solids in the system. This is fairly straightforward since the positions of the masses have been given as coordinates. We will call the solid at solid A, the solid at solid B, and the solid at solid C. The final column of the table contains the information for the system as a whole.
| Solid | A | B | C | System |
|---|---|---|---|---|
| Mass (kg) | 9 | 6 | ||
| -Coordinate | 0 | 0 | 0 | 0 |
| -Coordinate | 6 | 9 | 4 | 7 |
We know that the sum of the products of the masses and their respective -coordinate is equal to the product of the total mass and the -coordinate of the center of mass. Similarly, the sum of the products of the masses and their respective -coordinate is equal to the product of the total mass and the -coordinate of the center of mass. Therefore, we can now form two equations using the table:
In each element of the first equation, we are multiplying by zero, so it will simplify to . Therefore, only the second of these equations will be useful to us. Let us simplify this equation:
Now, we can move all the multiples of to one side and everything else to the other side as follows:
Finally, we divide both sides by 3 to reach our solution of
Method 2
In this question, we have been given the coordinates of three masses, as well as the coordinates of their center of mass. We can use the formula for the center of mass to relate these quantities, and then rearrange it to make the unknown mass, , the subject.
Let us first write the positions of the three masses as vectors: , , and . The position vector of the center of mass is .
The formula for the center of mass of a set of objects is
Let us substitute the values that we have been given:
Now, let us rearrange the equation to make the subject:
So, the mass of the third object is 1 kg.
In our final examples, let us look at problems where systems are presented geometrically.
Example 5: Finding the Center of Mass of a System of Three Masses Placed on the Sides of a Square
A square has side length . Three masses of 610 g are placed at , , and . Find the coordinates of the center of mass of the system.
Answer
There are two methods we could use to solve this problem. The first is to form and complete a table of the masses in the system and the second is to convert the positions of the masses to vector form and solve using the formula. We will be showing both methods here.
Method 1
For the first method, we need to find the coordinates of the three masses. We know that they lie on the corners of a square with side length . From the diagram, we can see that the mass is on the origin, the mass is on the -axis, and the mass is on the -axis. Therefore, we can say that the coordinates of mass are , the coordinates of mass are , and the coordinates of mass are .
We can now form a table containing the mass, -coordinate, and -coordinate of each of the masses in the system. The final column of the table contains the information for the system as a whole.
| Mass | System | |||
|---|---|---|---|---|
| Mass (g) | 610 | 610 | 610 | 1 830 |
| -Coordinate | 0 | 0 | ||
| -Coordinate | 0 | 0 |
Now, we can form two equations, one relating the masses to the -coordinate and the other relating the masses to the -coordinates, as follows:
Simplifying these equations, we obtain
Finally, if we divide by 610, we will see that
Hence the coordinates of the center of mass are
Method 2
In this question, we have been given the masses of three objects and their positions in terms of a constant, . This means that when we calculate the center of mass of the system, it is also going to be in terms of .
We can represent the positions of the objects at , , and as vectors. The object at has a position vector of ; the object at has a position vector of ; the object at has a position vector of .
We can use the formula to find the center of mass of the system. Let us substitute in the values:
A factor of 610 can be taken out of the numerator and the denominator on the right-hand side:
This is the position vector of the center of mass. We can also write it as a set of coordinates: .
Example 6: Finding the Center of Gravity of Three Equal Discrete Masses Placed on the Sides of a Triangle
A triangle , where , , , and and are the midpoints of and , respectively, is located in the first quadrant of a Cartesian plane such that is at the origin and the point is on the -axis. Three equal masses are placed at points , , and . Determine the coordinates of the center of gravity of the system.
Answer
There are two methods we could use to solve this problem. The first is to form and complete a table of the masses in the system and the second is to convert the positions of the masses to vector form and solve using the formula. We will be showing both methods here.
Method 1
Firstly, we need to work out the coordinates of the three masses. Mass is on the origin, so its coordinates are . Mass is on the midpoint of , and we know , so its coordinates are . It is a little more difficult to find the coordinates of mass . However, since is a right triangle and is at the midpoint of , the hypotenuse, that means its -coordinate will be half the width of the triangle and its -coordinate will be half of the height the triangle. The width of the triangle is 44 cm and the height is 33 cm, so its coordinates are .
We can now form a table containing the mass, -coordinate, and -coordinate of each of the masses in the system. The final column of the table contains the information for the system as a whole.
| Mass | System | |||
|---|---|---|---|---|
| Mass (g) | ||||
| -Coordinate | 0 | 0 | 22 | |
| -Coordinate | 0 |
Now, we can form two equations, one relating the masses to the -coordinate and another relating the masses to the -coordinate, as follows:
Since each term is a multiple of , we can cancel out this factor and simplify the terms in the two equations to obtain
Next, we divide both equations by 3:
Hence, we have found the coordinate of the center of mass. Our solution is
Method 2
Let us start by finding position vectors for the points , , and , where the masses are.
is at the origin, so its position vector is . is 33 cm away from along the -axis, so its position vector is . is at the midpoint between and , so its position vector is .
We can find the position vector of by adding the vectors and . is equal to , so the position vector of , equal to , is given by . Since , the position vector of is given by . is equal to and is equal to , so the position vector of is .
We can now use the formula to find the center of mass of the system in terms of . Let us substitute the values:
So, the center of mass of the system is at , or in coordinate form.
Let us finish by recapping some key points.
Key Points
- For a system of particles, where the particle in the system has position vector and mass , the position vector of the center of mass of the system, , is given by where is the total mass of all of the particles.
- Always consider the geometry of the system of particles as this can sometimes allow you to simplify the problem. If, for example, all of the particles lie on a horizontal, or vertical line then the center of mass of the system must also lie on the same line.
- It can be helpful to convert all points and coordinates into position vectors before attempting to find the center of mass.
