Question

Figure 13-28 shows three particles initially fixed in place, with Band Cidentical and positioned symmetrically about the yaxis, at distance dfrom A. (a) In what direction is the net gravitational force${\overrightarrow{\mathbf{F}}}_{\mathbf{n}\mathbf{e}\mathbf{t}}$ on A? (b) If we move Cdirectly away from the origin, does${\mathit{F}}_{\mathbf{n}\mathbf{e}\mathbf{t}}$change in direction? If so, how and what is the limit of the change?

Short answer

1. The net gravitational force${\overrightarrow{F}}_{net}$on A is along + y-axis.
2. The net gravitational force ${\overrightarrow{F}}_{net}$changes in direction if C is moved directly away from the origin until it coincides with B.

Step-by-step solution

The given data

The figure for three particle system initially fixed in place, with B and C identical and positioned symmetrically about y-axis at distance from A.

Understanding the concept of the gravitational force

The force of attraction between any two bodies is directly proportional to the product of their masses and is inversely proportional to the square of the distance between them, according to Newton's universal law of gravitation.Using the concept of the vector diagram, we can find the direction of the net force as per the given conditions.

Formula:

The Gravitational force of attraction between two bodies of masses M and m separated by distance R is, $F=\frac{GMm}{R^2}$ …(i)

(a) Calculation of the net gravitational force on A

Free body diagram:

From the above vector diagram, we can conclude that the direction of the net gravitational force ${\overrightarrow{F}}_{net}$on A is along +y direction.

(b) Calculation of the change in the net gravitational force

As C is moved away from the origin, the gravitational force ${\overrightarrow{F}}_{AC}$on A will decrease with distance. This is because force is inversely proportional to the square of a distance between the masses according to equation (i). So as C moves away, the net force would move towards B as force exerted by the particle at B on A would become greater than that of the particle at C.

At infinity, the value of force because the particle is at C would become zero and the net force would be the only force exerted by particle B on A.

Therefore, as particle C moves away to infinity, value of ${\overrightarrow{F}}_{net}$would decrease till it becomes equal to ${\overrightarrow{F}}_{AB}$.