Question

A coordinate system (in meters) is constructed on the surface of a pool table, and three objects are placed on the table as follows: a 2.0-kg object at the origin of the coordinate system, a 6.0-kg object at (0, 2.0), and a 11.0-kg object at (4.0, 0). Find the resultant gravitational force exerted by the other two objects on the object at the origin.

Answer 1

A coordinate system (in meters) is constructed on the surface of a pool table, and three objects are placed on the table as follows: a 7.0 kg object at the origin of the coordinate system, a 8.0 kg object at (0, 2.0), and a 17.0 kg object at (4.0, 0). Find the resultant gravitational force exerted by the other two objects on the object at the origin. Find the magnitude and direction of the force.   

Gravitational force between two objects F = G * m₁ * m₂ / r²

and r² = (x₂ - x₁)²  + (y₂  - y₁)²

then r₁ (between 7 and 17 kg) r₁²   = (4 - 0)²  + (0 - 0)²

= 16

F₁  = 6.67 * 10^-11 * 7 * 17 / 16

= 4.96 * 10^-10  N along + ve x axis

r₂²   = (0 - 0)²  + (2 - 0)²  

= 4

F₂   =   6.67 * 10^-11 * 7 * 8 / 16

= 2.33 * 10^-10  N along + ve y axis

Net force F = √(F₁² + F₂² + 2 * F₁ * F₂ * cos φ)

where   φ = 90⁰   or cos φ = 0

=> F = √(F₁² + F₂²)

= √{(4.96 * 10^-10)² + (2.33 * 10^-10)²}

= 5.48 * 10^-10  N

tan θ = F₂ / F₁

θ = tan^-1 {(2.33 * 10^-10) / (4.96 * 10^-10)}

= 25.16⁰

i.e. net force of 5.48 * 10^-10  N works at an angle   25.16⁰   above x axis.