Home
Class 12
PHYSICS
Calculate the force exerted by point mas...

Calculate the force exerted by point mass m on rod of uniformly distributed mass `M` and length l (Placed as shown in figure)
.

Text Solution

Verified by Experts

`because` Direction of force is changing at every element. We have to make components of force and then integrate.
Net vertical force `=0`
`dF` = force element `= (G.dM.m)/((x^(2)+a^(2))`
`dF_(h) = dF cos theta` = Force on element in horizontal direction `= (G.dM.m)/((x^(2)+a^(2)) cos theta`
`:.F_(h)=int(G.M.mcosthetadx)/(l(x^2+a^2))=(G.M.m)/l underset(-l//2)overset(l//2)int(costheta.dx)/((x^(2)+a^(2)))=(GMm)/(la^(2))underset(-l//2)overset(l//2)int(costheta.dx)/(sec^(2)theta)`
where `x =a` than `theta` then `dx = a sec^(2) theta d theta`
`=(GMm)/(la)[sintheta]_(l//2)^(l//2)tantheta=(x)/(a),thesintheta=(x)/(sqrt(x^(2)+a^(2)))`
=`(GMm)/(la)[(x)/(sqrt(x^(2)+a^(2)))]_(-l//2)^(l//2) =sqrt(GMml)/(lasqrt((l^(2))/(4)+a^(2)))=(GMm)/(asqrt(((l^(2))/(4)+a^(2))))`
.
Promotional Banner

Topper's Solved these Questions

  • GRAVITATION

    RESONANCE ENGLISH|Exercise EXAMPLE|2 Videos

  • GRAVITATION

    RESONANCE ENGLISH|Exercise EXERCISE-1 PART-1|9 Videos

  • GRAVITATION

    RESONANCE ENGLISH|Exercise HIGH LEVEL PROBLEMS|16 Videos

  • GEOMATRICAL OPTICS

    RESONANCE ENGLISH|Exercise Advance level Problems|35 Videos

  • NUCLEAR PHYSICS

    RESONANCE ENGLISH|Exercise Advanced level solutions|16 Videos

Similar Questions

Explore conceptually related problems

Calculate the moment of inertia of a uniform rod of mass M and length l about an axis passing through an end and perpendicular to the rod. The rod can be divided into a number of mass elements along the length of the rod.

Charge Q is uniformly distributed on a dielectric rod AB of length 2l . The potential at P shown in the figure is equal to

The moment of inertia of a thing uniform rod of length L and mass M , about axis CD (as shown in figure) is

Find a_(C) and alpha of the smooth rod of mass m and length l .

A straight of length L extends from x=a to x=L+a. the gravitational force it exerts on a point mass 'm' at x=0, if the mass per unit length of the rod is A+Bx^(2), is given by :

If a street light of mass M is suspended from the end of a uniform rod of length L in different possible patterns as shown in figure, then:

Two uniform solid of masses m_(1) and m_(2) and radii r_(1) and r_(2) respectively, are connected at the ends of a uniform rod of length l and mass m . Find the moment of inertia of the system about an axis perpendicular to the rod and passing through a point at a distance of a from the centre of mass of the rod as shown in figure.

A uniform rod of length l and mass m is hung from, strings of equal length from a ceiling as shown in figure. Determine the tensions in the strings?

Calculate the moment of inertia of a rod of mass M, and length l about an axis perpendicular to it passing through one of its ends.

Three point masses m,m and 2m are placed on the vertex of an equilateral triangle of side 'a' Another point mass 'm' is placed at the centroid of the triangle as shown in the figure. If the magnitude of net force on the point mass placed at centroid is (KGm^2)/(a^2) then find the value of K (G= universal gravitational constant )