Newton's Law of Gravitation — Question 4

Question 4

The force direction follows the displacement toward the source

A source $M$ is at $(0,0)$ and a test mass $m$ is at $(3a,4a)$, with $a>0$.

1. Find the Cartesian gravitational force on $m$.

2. Check its magnitude and find the opposite force on $M$.

Question 4 - Solution

The force direction follows the displacement toward the source

(a) Derivation and calculation

The displacement from source to particle is $\vec r=a(3\hat i+4\hat j)$, with $r=5a$. Therefore $$\vec F_m=-\frac{GMm}{r^3}\vec r =-\frac{GMm}{125a^2}(3\hat i+4\hat j).$$ $$\boxed{F_x=-\frac{3GMm}{125a^2},\quad F_y=-\frac{4GMm}{125a^2}}.$$

(b) Result and interpretation

The component norm gives $$|\vec F_m|=\frac{GMm}{125a^2}\sqrt{3^2+4^2} =\boxed{\frac{GMm}{25a^2}}=\frac{GMm}{(5a)^2}.$$ By action and reaction, $$\boxed{\vec F_M=-\vec F_m},\qquad \vec F_M+\vec F_m=\vec0.$$

Physical check

Both force components on the test mass are negative, pointing toward the origin.