Principle of Superposition of Gravitational Forces

Gravitational Force on a Point Mass due to Number of Other Point Masses

Statement

Principle of Superposition of Gravitational Forces:

The gravitational force on a point mass due to a number of other point masses around it is the vector sum of the gravitational forces acting on the point mass due to each of the other point masses separately.

Explanation

Consider a point mass m₁. Let this point mass be surrounded by other point masses m₂, m₃, m₄, m₅ and m₆. Let the distances of these masses from m₁ be r₁₂, r₁₃, r₁₄, r₁₅ and r₁₆ respectively.

Individual Gravitational Forces

Gravitational force on m₁ due to m₂:

$ \vec{F}_{12}=\frac{Gm_1m_2}{r_{12}^{2}}\hat{r}_{12} \qquad ...(i) $

where, \(\hat{r}_{12}\) is the unit vector representing the direction of \(\vec{F}_{12}\).

Gravitational force on m₁ due to m₃:

$ \vec{F}_{13}=\frac{Gm_1m_3}{r_{13}^{2}}\hat{r}_{13} \qquad ...(ii) $

Gravitational force on m₁ due to m₄:

$ \vec{F}_{14}=\frac{Gm_1m_4}{r_{14}^{2}}\hat{r}_{14} \qquad ...(iii) $

Gravitational force on m₁ due to m₅:

$ \vec{F}_{15}=\frac{Gm_1m_5}{r_{15}^{2}}\hat{r}_{15} \qquad ...(iv) $

Gravitational force on m₁ due to m₆:

$ \vec{F}_{16}=\frac{Gm_1m_6}{r_{16}^{2}}\hat{r}_{16} \qquad ...(v) $

Net Gravitational Force

Therefore, the net gravitational force acting on m₁ due to all the surrounding masses is:

$ \vec{F} = \vec{F}_{12} + \vec{F}_{13} + \vec{F}_{14} + \vec{F}_{15} + \vec{F}_{16} $

Substituting the values of individual forces, we get:

$ \vec{F} = \frac{Gm_1m_2}{r_{12}^{2}}\hat{r}_{12} + \frac{Gm_1m_3}{r_{13}^{2}}\hat{r}_{13} + \frac{Gm_1m_4}{r_{14}^{2}}\hat{r}_{14}$

 $+ \frac{Gm_1m_5}{r_{15}^{2}}\hat{r}_{15} + \frac{Gm_1m_6}{r_{16}^{2}}\hat{r}_{16} $

Thus, the resultant gravitational force on a particle due to several surrounding particles is equal to the vector sum of the individual gravitational forces acting on it.

Important Points

• Gravitational force is a vector quantity.
• The forces are added using vector addition.
• Each gravitational force acts independently.
• Newton's law of gravitation obeys Newton's third law of motion.

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FAQs

1. What is the principle of superposition of gravitational forces?

It states that the net gravitational force acting on a body due to several masses is equal to the vector sum of the individual forces exerted by those masses.

2. Why is vector addition used?

Because gravitational force has both magnitude and direction.

3. Does one mass affect the force due to another mass?

No. Each gravitational force acts independently.

4. Which law of motion is obeyed by Newton's law of gravitation?

Newton's third law of motion.

5. What is the SI unit of G?

N m² kg⁻².

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Quiz

1. The net gravitational force is the ______ of individual forces.

(a) Scalar sum
(b) Vector sum
(c) Product
(d) Difference

2. Gravitational force is a ______ quantity.

(a) Scalar
(b) Vector
(c) Dimensionless
(d) Constant

3. SI unit of G is:

(a) N m² kg⁻²
(b) N kg² m⁻²
(c) N m kg⁻²
(d) kg N m²

4. Gravitational forces between two masses are:

(a) Equal and same direction
(b) Unequal and opposite
(c) Equal and opposite
(d) Zero

5. Superposition principle uses:

(a) Scalar addition
(b) Vector addition
(c) Multiplication
(d) Division