The mass density of a spherical body is given | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The mass $M(r)$ enclosed within a sphere of radius $r$ $(r \leq R)$ is given by:$M(r) = \int_0^r \rho(r') 4\pi r'^2 dr' = \int_0^r \frac{k}{r'} 4\

Vedclass · Accepted Answer

Vedclass · Answer

(B) The mass $M(r)$ enclosed within a sphere of radius $r$ $(r \leq R)$ is given by:$M(r) = \int_0^r ho(r') 4\pi r'^2 dr' = \int_0^r \frac{k}{r'} 4\pi r'^2 dr' = 4\pi k \int_0^r r' dr' = 2\pi k r^2$.The acceleration $a$ of a test particle at distance $r$ is given by $a = \frac{GM(r)}{r^2}$.For $r \leq R$:$a = \frac{G(2\pi k r^2)}{r^2} = 2\pi G k = 	ext{constant}$.For $r > R$,the total mass $M = M(R) = 2\pi k R^2$ is constant.$a = \frac{GM}{r^2} = \frac{G(2\pi k R^2)}{r^2} \propto \frac{1}{r^2}$.Thus,the acceleration is constant for $r \leq R$ and decreases as $1/r^2$ for $r > R$. The graph corresponding to this behavior is option $(b)$.

The mass density of a spherical body is given by $\rho(r) = \frac{k}{r}$ for $r \leq R$ and $\rho(r) = 0$ for $r > R$,where $r$ is the distance from the centre. The correct graph that describes qualitatively the acceleration,$a$,of a test particle as a function of $r$ is

Similar Questions

Two particles of equal mass $(m)$ each move in a circle of radius $(r)$ under the action of their mutual gravitational attraction. Find the speed of each particle.

Four identical particles of equal masses $m = 1 \, kg$ are made to move along the circumference of a circle of radius $R = 1 \, m$ under the action of their own mutual gravitational attraction. The speed of each particle will be

Four particles,each of mass $M$ and equidistant from each other,move along a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle is

Identify the correct option.

Vedclass Test Series

Exam Paper Generator

Online Exam Module