Question

A uniform rod of mass M and L is considered. Find the gravitational field intensity at point P on perpendicular bisector?​

Answer 1

Answer:

Explanation:

A uniform rod of mass m and length l is taken. Find the gravitational field intensity at point P at distance d which is on the perpendicular bisector of the rod as shown in Fig.

Question from  Class 11 Maths Chapter Graviation

7.3 K+ views | 16.2 K+ people like this

Like

   

Share

Answer Text

4Gmd4d2+l2−−−−−−−√

2Gmdl2+4d2−−−−−−−√

22–√Gmdl2+4d2−−−−−−−√

none of the above

Answer :

B

Solution :

Consider element of the rod of width x as shown in Fig.

Mass of element, dm=mldx

Gravitaitonal field intensity at P due to this element is

dI=Gdmx2+d2=G(x2+d2)×mldx along PC

In δOPC,x=dtanθ and dx=dsec2θdθ

∴x2+d2=d2(tan2θ+1)=d2sec2θ

∴dI=Gmldsec2θdθd2sec2θ=Gmlddθ

resolving dI→ into two rectangular components we have dIcosθ acts along PO and dIsinθ acts perpendicular to PO.

If we find the gravitaitonal field due to other elements of the rod at P and resolve them into two rectangular componets, we note that the components of gravitational field intensity perpendicular to OP will cancel out and components of gravitational intensity along PO will be added up. Therefore, total gravitational intensity at P due to entire rod is

I=∫+α−αdIcosθ=∫+α−αGmldcosθdθ

=Gmld[2sinα]=2Gmld×l/2(l/2)2+d2−−−−−−−−−√

=2Gmdl2+4d2−−−−−−−√