question_answer

A straight rod of length L extends from \[x=a\]to\[x=L+a\]. Find the gravitational force it exerts on a point mass m at x = 0 if the linear density of rod\[\mu =A+B{{x}^{2}}\].                                              

A) \[Gm\left[ \frac{A}{a}+BL \right]\]

B) \[Gm\left[ A\left( \frac{1}{a}-\frac{1}{a+L} \right)+BL \right]\]

C) \[Gm\left[ BL+\frac{A}{a+L} \right]\]

D) \[Gm\left[ BL+\frac{A}{a} \right]\]

Correct Answer: B

Solution :

[b] \[\because dF=\frac{GM\left( \mu dx \right)}{{{x}^{2}}}\] \[F=GM\int{\left( A+Bx \right)\frac{dx}{x}}\] \[F=GM\left[ A\left( \frac{1}{a}-\frac{1}{a+L} \right)+BL \right]\]