Three uniform rods, each of mass $M$ and length $l$ , are connected to form an equilateral triangle in a gravity free space. Another small body of mass $m$ is kept at the centroid. Find the minimum velocity $v$ to be given to mass $m$ so that it escapes the gravitational pull of the triangle.

Open in App

Solution

The gravitational potential energy between the point mass $m$ and the elementary segment is given as
$d U = - \frac{G m (d m)}{r} = - G m {(\frac{M}{l}) d x} / \sqrt{a^{2} + x^{2}}$
$= - \frac{G M m}{l} \frac{d x}{\sqrt{a^{2} + x^{2}}}$
The total gravitational potential energy associated with the system, $U = 3 \int d U$
$= - (\frac{3 G M m}{l}) 2 \int_{0}^{l / 2} \frac{d x}{\sqrt{a^{2} + x^{2}}}$
$= - \frac{6 G M m}{l} {[l n ∣ ∣ ∣ \frac{x}{a} + \frac{\sqrt{a^{2} + x^{2}}}{a} ∣ ∣ ∣]}_{0}^{l / 2}$
$\Rightarrow U = - \frac{6 G M m}{l} l n ∣ ∣ ∣ ∣ \frac{l / 2 + \sqrt{a^{2} (l / 2)^{2}}}{a} ∣ ∣ ∣ ∣$
$U = - \frac{6 G M m}{l} l n ∣ ∣ ∣ \frac{l + \sqrt{l^{2} + 4 a^{2}}}{2 a} ∣ ∣ ∣$
For minimum velocity $| U | = \frac{1}{2} m v^{2}$
$∴ v = {[\frac{12 G M}{l} . l n ∣ ∣ ∣ \frac{l + \sqrt{l^{2} + 4 a^{2}}}{2 a} ∣ ∣ ∣]}^{1 / 2}$ Put, $a = \frac{l}{3} sin 60^{\circ} = \frac{l}{2 \sqrt{3}}$
$∴ v = {[\frac{12 G M}{l} . l n (2 + \sqrt{3})]}^{1 / 2}$ .

Suggest Corrections

Similar questions

Q. Three mass points each of mass m are placed at the vertices of an equilateral triangle of side l. What is the gravitational field and potential due to three masses at the centroid of the triangle?

Find the moment of inertia of 3 rods each of mass $M$ and length $L$ , connected to form an equilateral triangle about $M N$

Q. Three identical thin rods, each of mass m and length l are joined to form an equilateral triangle. Find the moment of inertia of the triangle about one of its sides.

Q. Three particles each of mass m are kept at verticles of an equilateral triangle of side L. The gravitational field at centre due to these particle is:

Three particles, each of mass m, are placed at the vertices of an equilateral triangle of side $a$ . The net gravitational force on a body of mass $M$ at the centroid of the triangle is

Join BYJU'S Learning Program

Three uniform rods, each of mass M and length l, are connected to form an equilateral triangle in a gravity free space. Another small body of mass m is kept at the centroid. Find the minimum velocity v to be given to mass m so that it escapes the gravitational pull of the triangle.

Three uniform rods, each of mass $M$ and length $l$ , are connected to form an equilateral triangle in a gravity free space. Another small body of mass $m$ is kept at the centroid. Find the minimum velocity $v$ to be given to mass $m$ so that it escapes the gravitational pull of the triangle.