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Q.

A straight rod of length L extends from $x = a$ to $x = L + a$ . The linear mass density of the rod varies with x-coordinate is $λ = A + {Bx}^{2}$ . The gravitational force experienced by a point mass m at $x = 0$ is..

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a

$Gm [\frac{A}{a} + BL]$

b

$Gm [BL + \frac{A}{a + L}]$

c

$Gm [A [\frac{1}{a} - \frac{1}{a + L}] + BL]$

d

$Gm [\frac{BL}{a} + \frac{A}{a^{2}}]$

answer is C.

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Detailed Solution

$dF = \frac{G \cdot m \cdot dm}{x^{2}}; dF = \frac{Gm λ dx}{x^{2}}$

Question Image

$F = \int_{a}^{a + L} d F = \int_{a}^{a + L} \frac{G m λ d x}{x^{2}}$

$F = Gm \int_{a}^{a + L} [\frac{A + {Bxx}^{2}}{x^{2}}] dx$

$F = Gm [A [\frac{1}{a} - \frac{1}{a + L}] + BL]$

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A straight rod of length L extends from x=a to x=L+a. The linear mass density of the rod varies with x-coordinate is λ=A+Bx2 . The gravitational force experienced by a point mass m at x=0 is..