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Four particles of mass m,2m,3m,4m kept in a sequence at the corner of a square of side "a" magnitude of gravitational force acting on a particle of mass m , placed at the centre of the square will be?
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Four particles of mass m,2m,3m,4m kept in a sequence at the corner of ...
Gravitational Force on a Particle at the Center of a Square
To find the magnitude of the gravitational force acting on a particle of mass m placed at the center of a square, we need to consider the gravitational forces exerted by each of the four particles in the sequence.
Gravitational Force Calculation
The gravitational force between two objects is given by Newton's Law of Universal Gravitation:
F = G * (m1 * m2) / r^2
Where:
- F is the gravitational force between the two objects
- G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2)
- m1 and m2 are the masses of the two objects
- r is the distance between the centers of the two objects
Gravitational Forces from Each Particle
Let's consider each particle in the sequence and calculate the gravitational force exerted by it on the particle at the center of the square.
1. Particle of mass m:
The distance between the particle at the center and the particle of mass m is the length of the side of the square, which is "a". Therefore, the gravitational force exerted by this particle is:
F1 = G * (m * m) / a^2
2. Particle of mass 2m:
The distance between the particle at the center and the particle of mass 2m is the diagonal of the square, which can be calculated using the Pythagorean theorem. The length of the diagonal is √2a. Therefore, the gravitational force exerted by this particle is:
F2 = G * (m * 2m) / (√2a)^2 = G * (2m^2) / 2a^2 = G * m^2 / a^2
3. Particle of mass 3m:
The distance between the particle at the center and the particle of mass 3m is also the diagonal of the square, which is √2a. Therefore, the gravitational force exerted by this particle is:
F3 = G * (m * 3m) / (√2a)^2 = G * (3m^2) / 2a^2
4. Particle of mass 4m:
The distance between the particle at the center and the particle of mass 4m is the diagonal of the square, which is √2a. Therefore, the gravitational force exerted by this particle is:
F4 = G * (m * 4m) / (√2a)^2 = G * (4m^2) / 2a^2 = 2G * m^2 / a^2
Net Gravitational Force
To find the net gravitational force on the particle at the center, we need to sum up the forces from each particle:
Net Force = F1 + F2 + F3 + F4
= G * (m^2 / a^2) + G * (m^2 / a^2) + G * (3m^2 / 2a^2) + 2G * (m^2 / a^2)
= (5G * m^2 / a^2) + (3G * m^2 / 2a
To find the magnitude of the gravitational force acting on a particle of mass m placed at the center of a square, we need to consider the gravitational forces exerted by each of the four particles in the sequence.
Gravitational Force Calculation
The gravitational force between two objects is given by Newton's Law of Universal Gravitation:
F = G * (m1 * m2) / r^2
Where:
- F is the gravitational force between the two objects
- G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2)
- m1 and m2 are the masses of the two objects
- r is the distance between the centers of the two objects
Gravitational Forces from Each Particle
Let's consider each particle in the sequence and calculate the gravitational force exerted by it on the particle at the center of the square.
1. Particle of mass m:
The distance between the particle at the center and the particle of mass m is the length of the side of the square, which is "a". Therefore, the gravitational force exerted by this particle is:
F1 = G * (m * m) / a^2
2. Particle of mass 2m:
The distance between the particle at the center and the particle of mass 2m is the diagonal of the square, which can be calculated using the Pythagorean theorem. The length of the diagonal is √2a. Therefore, the gravitational force exerted by this particle is:
F2 = G * (m * 2m) / (√2a)^2 = G * (2m^2) / 2a^2 = G * m^2 / a^2
3. Particle of mass 3m:
The distance between the particle at the center and the particle of mass 3m is also the diagonal of the square, which is √2a. Therefore, the gravitational force exerted by this particle is:
F3 = G * (m * 3m) / (√2a)^2 = G * (3m^2) / 2a^2
4. Particle of mass 4m:
The distance between the particle at the center and the particle of mass 4m is the diagonal of the square, which is √2a. Therefore, the gravitational force exerted by this particle is:
F4 = G * (m * 4m) / (√2a)^2 = G * (4m^2) / 2a^2 = 2G * m^2 / a^2
Net Gravitational Force
To find the net gravitational force on the particle at the center, we need to sum up the forces from each particle:
Net Force = F1 + F2 + F3 + F4
= G * (m^2 / a^2) + G * (m^2 / a^2) + G * (3m^2 / 2a^2) + 2G * (m^2 / a^2)
= (5G * m^2 / a^2) + (3G * m^2 / 2a
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