Question

The gravitational field intensity $\overrightarrow{E}$ of earth at any point is defined as the gravitational force per unit mass at that point. It varies from place to place. The variation is shown in column II with position $\overrightarrow{r}$ vs the $\overrightarrow{E}$ in the form of graphs. The variation of $\overrightarrow{r}$ is given in column I. Choose the correct form of graphs for the corresponding variations of $\overrightarrow{r}.$

• A. $A\to 1;B\to 2;C\to 3;D\to 4$
• B. $A\to 4;B\to 3;C\to 2;D\to 1;$
• C. $A\to 4;B\to 1;C\to 3;D\to 2$
• D. $A\to 3;B\to 2;C\to 1;D\to 4$

Answer 1

The correct option is D $A\to 3;B\to 2;C\to 1;D\to 4$

Gravitational field intensity at any point outside due to earth is given by
$E=g=\frac{GM}{(R+h{)}^2}=\frac{GM}{r^2}$
Where, h is height from surface or $E\propto \frac{1}{r^2}but-ve$ in sign,
Graph is parabolic.
Inside earth g decreases as r increases form surface of earth.
$E=g(1-\frac{r}{R})=g-\frac{g}{R}r$
E is straight line decreasing to 0 at centre of earth and then increases in magnitude but -ve as directed towards the centre always.
At any point outside $E\propto \frac{1}{r^2}$ (parabolic graph)
Inside earth E $\propto$ r (linear graph)
In hallow sphere inside E = 0 graph along r-axis; outside as if whole mass lies at centre $E\propto \frac{1}{r^2}$, decreasing parabolic graph after sudden rise at surface of earth.