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If three particles of masses 2kg,1kg and 3kg are placed at corners of an equilateral triangle of perimeter 6m then the distance of center of mass which is at origin of particles from 1kg mass is (approximately)(assume 2kg on x-axis)?
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If three particles of masses 2kg,1kg and 3kg are placed at corners of ...
Given:
- Mass of particle 1 (m1) = 2 kg
- Mass of particle 2 (m2) = 1 kg
- Mass of particle 3 (m3) = 3 kg
- Perimeter of equilateral triangle (P) = 6 m
To find:
- Distance of center of mass from the 1 kg mass
Solution:
Step 1: Finding the coordinates of the particles
- Let's assume the equilateral triangle is placed in the xy-plane, with the 2 kg mass at the origin (0, 0).
- The coordinates of the 1 kg mass (m2) can be found using trigonometry. Since it is an equilateral triangle, we can divide the perimeter equally between the three sides, giving us a side length (s) of 2 m.
- Using the cosine rule, we can find the coordinates of the 1 kg mass:
- x-coordinate of m2 = s * cos(60°) = 2 * 0.5 = 1 m
- y-coordinate of m2 = s * sin(60°) = 2 * (√3/2) = √3 m
- The coordinates of m2 are (1, √3).
Step 2: Finding the center of mass
- The center of mass (CM) can be found using the formula:
- CMx = (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3)
- CMy = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3)
- Plugging in the values:
- CMx = (2 * 0 + 1 * 1 + 3 * 0) / (2 + 1 + 3) = 1 / 6
- CMy = (2 * 0 + 1 * √3 + 3 * 0) / (2 + 1 + 3) = √3 / 6
- The coordinates of the center of mass are (1/6, √3/6).
Step 3: Finding the distance from the center of mass to the 1 kg mass
- The distance between two points (x1, y1) and (x2, y2) can be found using the distance formula:
- Distance = √((x2 - x1)^2 + (y2 - y1)^2)
- Plugging in the values:
- Distance = √((1/6 - 1)^2 + (√3/6 - √3)^2)
- Distance = √((1/6 - 6/6)^2 + (√3/6 - 6√3/6)^2)
- Distance = √((-5/6)^2 + (-5√3/6)^2)
- Distance = √(25/36 + 75/36)
- Distance = √100/36
- Distance = 10/6
- Distance = 5/3
- Mass of particle 1 (m1) = 2 kg
- Mass of particle 2 (m2) = 1 kg
- Mass of particle 3 (m3) = 3 kg
- Perimeter of equilateral triangle (P) = 6 m
To find:
- Distance of center of mass from the 1 kg mass
Solution:
Step 1: Finding the coordinates of the particles
- Let's assume the equilateral triangle is placed in the xy-plane, with the 2 kg mass at the origin (0, 0).
- The coordinates of the 1 kg mass (m2) can be found using trigonometry. Since it is an equilateral triangle, we can divide the perimeter equally between the three sides, giving us a side length (s) of 2 m.
- Using the cosine rule, we can find the coordinates of the 1 kg mass:
- x-coordinate of m2 = s * cos(60°) = 2 * 0.5 = 1 m
- y-coordinate of m2 = s * sin(60°) = 2 * (√3/2) = √3 m
- The coordinates of m2 are (1, √3).
Step 2: Finding the center of mass
- The center of mass (CM) can be found using the formula:
- CMx = (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3)
- CMy = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3)
- Plugging in the values:
- CMx = (2 * 0 + 1 * 1 + 3 * 0) / (2 + 1 + 3) = 1 / 6
- CMy = (2 * 0 + 1 * √3 + 3 * 0) / (2 + 1 + 3) = √3 / 6
- The coordinates of the center of mass are (1/6, √3/6).
Step 3: Finding the distance from the center of mass to the 1 kg mass
- The distance between two points (x1, y1) and (x2, y2) can be found using the distance formula:
- Distance = √((x2 - x1)^2 + (y2 - y1)^2)
- Plugging in the values:
- Distance = √((1/6 - 1)^2 + (√3/6 - √3)^2)
- Distance = √((1/6 - 6/6)^2 + (√3/6 - 6√3/6)^2)
- Distance = √((-5/6)^2 + (-5√3/6)^2)
- Distance = √(25/36 + 75/36)
- Distance = √100/36
- Distance = 10/6
- Distance = 5/3
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If three particles of masses 2kg,1kg and 3kg are placed at corners of an equilateral triangle of perimeter 6m then the distance of center of mass which is at origin of particles from 1kg mass is (approximately)(assume 2kg on x-axis)?
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If three particles of masses 2kg,1kg and 3kg are placed at corners of an equilateral triangle of perimeter 6m then the distance of center of mass which is at origin of particles from 1kg mass is (approximately)(assume 2kg on x-axis)? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about If three particles of masses 2kg,1kg and 3kg are placed at corners of an equilateral triangle of perimeter 6m then the distance of center of mass which is at origin of particles from 1kg mass is (approximately)(assume 2kg on x-axis)? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If three particles of masses 2kg,1kg and 3kg are placed at corners of an equilateral triangle of perimeter 6m then the distance of center of mass which is at origin of particles from 1kg mass is (approximately)(assume 2kg on x-axis)?.
If three particles of masses 2kg,1kg and 3kg are placed at corners of an equilateral triangle of perimeter 6m then the distance of center of mass which is at origin of particles from 1kg mass is (approximately)(assume 2kg on x-axis)? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about If three particles of masses 2kg,1kg and 3kg are placed at corners of an equilateral triangle of perimeter 6m then the distance of center of mass which is at origin of particles from 1kg mass is (approximately)(assume 2kg on x-axis)? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If three particles of masses 2kg,1kg and 3kg are placed at corners of an equilateral triangle of perimeter 6m then the distance of center of mass which is at origin of particles from 1kg mass is (approximately)(assume 2kg on x-axis)?.
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