Consider a square ABCD of side 10cm and total mass 10kg. Let O be the center of the sqaure. We cut a square of corners O and C.
Find the central mass of the obtained shape after removing the small cutten sqaure.
I would appreciate your help. thank you
To find the central mass of the obtained shape after removing the small cut square, we need to find the coordinates of the center of mass for each component and then calculate the weighted average of their masses.
1. Let's start by finding the coordinates of the center of mass (COM) for the original square. Since the square is symmetric with respect to both the x-axis and y-axis, the center of mass will coincide with the center of the square O.
Therefore, COM for the original square = coordinates of O = (0, 0).
2. Next, let's determine the coordinates of the center of mass for the small cut square. Since we have removed the corners O and C, the new square will be divided into four quadrants.
- Quadrant 1: The mass in this quadrant is concentrated at point C. Since it is one-fourth of the total square, its mass will be one-fourth of the total mass, i.e., (10kg / 4) = 2.5kg. The coordinates of C are (5 cm, -5 cm). So the COM for this quadrant is (5 cm, -5 cm) multiplied by the mass, which is (2.5 kg).
- Quadrant 2: The mass in this quadrant is concentrated at point D. Similar to Quadrant 1, since it is one-fourth of the total square, its mass will also be one-fourth of the total mass, i.e., (10kg / 4) = 2.5kg. The coordinates of D are (5 cm, 5 cm). So the COM for this quadrant is (5 cm, 5 cm) multiplied by the mass, which is (2.5 kg).
- Quadrant 3: The mass in this quadrant is concentrated at point A. Similar to the previous quadrants, since it is one-fourth of the total square, its mass will be one-fourth of the total mass, i.e., (10kg / 4) = 2.5kg. The coordinates of A are (-5 cm, 5 cm). So the COM for this quadrant is (-5 cm, 5 cm) multiplied by the mass, which is (2.5 kg).
- Quadrant 4: The mass in this quadrant is concentrated at point B. Similar to the previous quadrants, since it is one-fourth of the total square, its mass will also be one-fourth of the total mass, i.e., (10kg / 4) = 2.5kg. The coordinates of B are (-5 cm, -5 cm). So the COM for this quadrant is (-5 cm, -5 cm) multiplied by the mass, which is (2.5 kg).
3. Finally, we need to find the center of mass of the obtained shape by calculating the weighted average of the coordinates of each component.
COM of the obtained shape = [(COM of original square) x (mass of original square) + (COM of quadrant 1) + (COM of quadrant 2) + (COM of quadrant 3) + (COM of quadrant 4)] / (total mass of the obtained shape).
Plugging in the values:
COM of the obtained shape = [(0, 0) x (10kg) + (5 cm, -5 cm) x (2.5 kg) + (5 cm, 5 cm) x (2.5 kg) + (-5 cm, 5 cm) x (2.5 kg) + (-5 cm, -5 cm) x (2.5 kg)] / (10 kg + 2.5 kg + 2.5 kg + 2.5 kg + 2.5 kg).
Simplifying the equation:
COM of the obtained shape = [(0, 0) + (12.5 cm, -12.5 cm) + (12.5 cm, 12.5 cm) + (-12.5 cm, 12.5 cm) + (-12.5 cm, -12.5 cm)] / (10 kg + 2.5 kg + 2.5 kg + 2.5 kg + 2.5 kg).
COM of the obtained shape = (0 cm, 0 cm).
Therefore, the center of mass of the obtained shape, after removing the small cut square, is located at the origin (0 cm, 0 cm).