three masses 3kg,4kg,5kg are located at corners of an equilateral triangle of side 1m.locate the centre of mass of the system
The x coordinates of masses 3kg,4kg,5kg are 0,m,m/2 respectively.
So the x coordinates of COM is
3×0+4×m+5×(m/2) / (3+4+5) (whole upon)
=13m/24
To locate the center of mass of the system, we need to find the position vector of each mass relative to the origin. Given that the masses are located at the corners of an equilateral triangle with a side length of 1m, we can use the vectors along the sides of the triangle to find the position vectors.
Let's assume the origin is at one of the corners of the triangle, and we label the points as A, B, and C. The mass of 3kg is at point A, the mass of 4kg is at point B, and the mass of 5kg is at point C.
To find the position vector of mass A relative to the origin, we can use the coordinates (1, 0) since it is located 1m away horizontally from the origin.
For mass B, we can use the coordinates (-0.5, √3/2), since it is located halfway horizontally and √3/2 vertically from the origin.
For mass C, we can use the coordinates (-0.5, -√3/2), since it is located halfway horizontally and -√3/2 vertically from the origin.
Now, we can calculate the center of mass using the formula:
Center of mass = (m₁ * d₁ + m₂ * d₂ + m₃ * d₃) / (m₁ + m₂ + m₃),
where m₁, m₂, and m₃ are the masses, and d₁, d₂, and d₃ are their respective position vectors.
Using the given masses and position vectors, the center of mass can be calculated as follows:
Center of mass = ((3kg * (1, 0)) + (4kg * (-0.5, √3/2)) + (5kg * (-0.5, -√3/2))) / (3kg + 4kg + 5kg)
= (3kg * (1, 0) + 4kg * (-0.5, √3/2) + 5kg * (-0.5, -√3/2)) / 12kg
= (3kg * (1, 0) + 4kg * (-0.5, √3/2) + 5kg * (-0.5, -√3/2)) / 12kg
= (3kg * (1, 0) - 2kg * (0.5, -√3/2) - 2.5kg * (0.5, √3/2)) / 12kg
= (3kg * (1, 0) - 1kg * (1, -√3) - 1kg * (1, √3)) / 12kg
= (2kg, 0) / 12kg
= (1/6, 0).
Therefore, the center of mass of the system is located at (1/6, 0).
To locate the center of mass of the system, you need to find the coordinates of the center of mass in terms of the positions and masses of the three masses. Here's how you can approach this problem:
Step 1: Assign coordinates to each mass
Let's assume the mass of 3kg is located at point A, the mass of 4kg is located at point B, and the mass of 5kg is located at point C. Assign coordinates to each point as follows:
A: (0, 0)
B: (1, 0)
C: (0.5, √3/2)
Step 2: Calculate the x-coordinate of the center of mass
The x-coordinate of the center of mass, denoted as X_cm, can be calculated using the formula:
X_cm = (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3)
where m1, m2, m3 are the masses of the three masses, and x1, x2, x3 are the corresponding x-coordinates.
For our system, the x-coordinate can be calculated as:
X_cm = (3 * 0 + 4 * 1 + 5 * 0.5) / (3 + 4 + 5) = 0.65
Step 3: Calculate the y-coordinate of the center of mass
The y-coordinate of the center of mass, denoted as Y_cm, can be calculated using the formula:
Y_cm = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3)
where m1, m2, m3 are the masses of the three masses, and y1, y2, y3 are the corresponding y-coordinates.
For our system, the y-coordinate can be calculated as:
Y_cm = (3 * 0 + 4 * 0 + 5 * √3/2) / (3 + 4 + 5) ≈ 0.81
Therefore, the center of mass of the system is located at approximately (0.65, 0.81).