Question

Four point masses 2kg, 4kg, 6kg and are placed at the corners of Square ABCD of 2cm long respectively. Find the Position of centre of mass of the system from the corner​

Answer 1

To find the position of the center of mass of the system, we need to first calculate the coordinates of the center of mass along both the x-axis and y-axis.

Let's begin by finding the coordinates of the center of mass along the x-axis:

We can use the formula:

x_cm = (m1x1 + m2x2 + m3x3 + m4x4) / (m1 + m2 + m3 + m4)

where m1, m2, m3, and m4 are the masses of the point masses at each corner, and x1, x2, x3, and x4 are the x-coordinates of each point mass.

In this case, we have:

m1 = 2kg, x1 = 0cm

m2 = 4kg, x2 = 2cm

m3 = 6kg, x3 = 2cm

m4 = 4kg, x4 = 0cm

Substituting these values into the formula, we get:

x_cm = (2kg x 0cm + 4kg x 2cm + 6kg x 2cm + 4kg x 0cm) / (2kg + 4kg + 6kg + 4kg)

x_cm = 24 / 16

x_cm = 1.5cm

Therefore, the x-coordinate of the center of mass is 1.5cm.

Now let's find the coordinates of the center of mass along the y-axis:

We can use the formula:

y_cm = (m1y1 + m2y2 + m3y3 + m4y4) / (m1 + m2 + m3 + m4)

where m1, m2, m3, and m4 are the masses of the point masses at each corner, and y1, y2, y3, and y4 are the y-coordinates of each point mass.

In this case, we have:

m1 = 2kg, y1 = 0cm

m2 = 4kg, y2 = 0cm

m3 = 6kg, y3 = 2cm

m4 = 4kg, y4 = 2cm

Substituting these values into the formula, we get:

y_cm = (2kg x 0cm + 4kg x 0cm + 6kg x 2cm + 4kg x 2cm) / (2kg + 4kg + 6kg + 4kg)

y_cm = 20 / 16

y_cm = 1.25cm

Therefore, the y-coordinate of the center of mass is 1.25cm.

So the center of mass of the system is located at the point (1.5cm, 1.25cm) from the corner.

Answer 2

We can find the position of the center of mass of the system by using the formula:

Xcm = (m1x1 + m2x2 + m3x3 + m4x4) / (m1 + m2 + m3 + m4)

where Xcm is the x-coordinate of the center of mass, m1, m2, m3, and m4 are the masses of the point masses, and x1, x2, x3, and x4 are their respective x-coordinates.

Similarly, we can find the y-coordinate of the center of mass using the formula:

Ycm = (m1y1 + m2y2 + m3y3 + m4y4) / (m1 + m2 + m3 + m4)

where Ycm is the y-coordinate of the center of mass, m1, m2, m3, and m4 are the masses of the point masses, and y1, y2, y3, and y4 are their respective y-coordinates.

Let's label the masses and coordinates as follows:

m1 = 2kg, x1 = 0cm, y1 = 0cm
m2 = 4kg, x2 = 2cm, y2 = 0cm
m3 = 6kg, x3 = 2cm, y3 = 2cm
m4 = 8kg, x4 = 0cm, y4 = 2cm

Substituting these values into the formulas, we get:

Xcm = (2kg x 0cm + 4kg x 2cm + 6kg x 2cm + 8kg x 0cm) / (2kg + 4kg + 6kg + 8kg) = 2cm

Ycm = (2kg x 0cm + 4kg x 0cm + 6kg x 2cm + 8kg x 2cm) / (2kg + 4kg + 6kg + 8kg) = 1cm

Therefore, the center of mass of the system is located 2cm from corner A in the x-direction and 1cm from corner A in the y-direction.