Gravitation Test

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Gravitation Test - 34

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Question 1
1 / -0

A man weighs $$80 kg$$ on the surface of earth of radius $$R$$. At what height above the surface of earth his weight will be $$40 kg$$?

A
$$\dfrac { R }{ 2 } $$

B
$$\sqrt { 2 } { R }$$

C
$$\left( \sqrt { 2 } -{ 1 } \right) { R }$$

D
$$\left( \sqrt { 2 } +{ 1 } \right) { R }$$

Solution

Let $$r$$ be the distance from center of earth where it weighs 40 kg.
$$W_{surface}=\dfrac{GMm}{R^2}$$
$$W_{r}=\dfrac{GMm}{r^2}$$
Since $$W_{r}=\dfrac{1}{2}W_{p}$$,
$$r=\sqrt{2}R$$.
Hence the height of this point$$=r-R=\sqrt{2}R-R=(\sqrt{2}-1)R$$
Question 2
1 / -0

Let $$V$$ and $$E$$ denote the gravitational potential and gravitational field at a point. It is possible to have

A
$$\displaystyle V=0$$ and $$E=0$$

B
$$\displaystyle V=0$$ and $$E \neq 0$$

C
$$\displaystyle V\neq 0$$ and $$E = 0$$

D
All of the above

Solution

A) At $$\infty$$ both V and E are zero.
B) Let, $$V_ \infty = \dfrac{GM}{R}$$ for an unit mass.
So, $$V_R = 0$$ i.e. at the radius $$R$$ of solid sphere(mass $$M$$) and $$ E_R = \dfrac{GM}{R^2} $$
C) Inside a spherical shell V =$$\dfrac{-GM}{R}$$ and E =0
Thus, all the above(D) is correct.
Question 3
1 / -0

The escape velocity of a projectile from the earth is approximately

A
$$\displaystyle 7$$ $$km/sec$$

B
$$\displaystyle 112$$ $$km/sec$$

C
$$\displaystyle 11.2$$ $$km/sec$$

D
$$\displaystyle 1.1$$ $$km/sec$$

Solution

The escape velocity of projectile from earth is
$$\displaystyle v_{e}=\sqrt {2g{R_e}}$$, where $$\displaystyle R_{e}$$ is radius of earth
since $$\displaystyle g=9.8m/sec^2, R_{e}=6.4\times10^6$$ metre
$$\displaystyle \Rightarrow v_{e}=11.2km/sec$$
Question 4
1 / -0

The escape velocity of an object projected from the surface of a given planet is independent of

A
Radius of the planet

B
The direction of projection

C
The mass of the planet

D
None of these

Solution

The escape velocity of an object from any planet is
$$\displaystyle v_{escape}=\sqrt {2gR} = \sqrt {2GM/R}$$
Where $$R$$ & $$M$$ are the radius & mass of the planet
Question 5
1 / -0

There is no atmosphere on the moon because.

A
It is closer to the earth

B
It revolves round the earth

C
It gets light from the sun

D
The escape velocity of gas molecules is lesser than their root mean square velocity

Solution

There is no atmosphere at moon because escape velocity is less than the root mean square velocity of the molecules at moon. Hence all molecules escape.
Question 6
1 / -0

The weight of an object in the coal mine, sea level and at the top of the mountain, are respectively $$W_1$$, $$W_2$$ and $$W_3$$ then

A
$$\displaystyle W_1 < W_2 > W_3$$

B
$$\displaystyle W_1 = W_2 = W_3$$

C
$$\displaystyle W_1 < W_2 < W_3$$

D
$$\displaystyle W_1 > W_2 > W_3$$

Solution

At the surface of earth, the value of $$g=9.8m/sec^2$$. If we go towards the centre of earth or we go above the surface of earth, then in both the cases the value of $$g$$ decreases.
Hence $$\displaystyle W_1=mg_{mine}, W_2=mg_{sea level}, W_3=mg_{mountain}$$
So $$\displaystyle W_1<W_2>W_3$$ ($$g$$ at the sea level $$=g$$ at the surface of earth)
Question 7
1 / -0

If $$g$$ is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass $$m$$ raised from the surface of the earth to a height equal to the radius $$R$$ of the earth is

A
$$\cfrac { 1 }{ 2 } mgR$$

B
$$2mgR$$

C
$$\cfrac { 1 }{ 4 } mgR$$

D
$$mgR$$

Solution

Gain in potential energy
$$=(-\dfrac{GMm}{2R})-(-\dfrac{GMm}{R})$$
$$=\dfrac{GMm}{R}-\dfrac{GMm}{2R}$$
$$=\dfrac{GMm}{2R}$$
$$=\dfrac{mgR^2}{2R} As, GM=gR^2$$
$$=\dfrac{1}{2}mgR$$
Question 8
1 / -0

The weight of a body at the centre of the earth is

A
Zero

B
Infinite

C
Same as on the surface on earth

D
None of these

Solution

Hint:
The magnitude of the gravitational force on a body equals the weight of the body.

Solution:
Step 1:
Concept used:
The force that pulls a body towards the center of the earth is its weight.
$$W =mg$$ .......(1)

where 'g' is the acceleration due to gravity,
W is the weight of a body of mass 'm'.

Step2:
A test mass always experiences an acceleration towards the center of the earth so it always tries to go towards the center of the earth. therefore, if the test mas is placed at the center of the earth then it will not feel any acceleration towards any other point, therefore the acceleration at the center of the earth is zero.

Calculating the weight of a body at the center of the earth:
The force exerted by any portion of the Earth at its center will be neutralized by the force exerted by the opposite portion. When the value of g is substituted in equation (1) as zero, the total weight will be zero.
Question 9
1 / -0

The largest and the shortest distance of the earth from the sun is $$r_1$$ and $$r_2$$. Its distance from the sun when it is at perpendicular to the major axis of the orbit drawn from the sun:

A
$$\displaystyle (r_1+r_2)/4$$

B
$$\displaystyle (r_1+r_2)/(r_1-r_2)$$

C
$$\displaystyle 2r_1r_2/(r_1+r_2)$$

D
$$\displaystyle (r_1+r_2)/3$$

Solution

Using the ellipse property is :
$$2/R=1/r_1 + 1/r_2$$
$$R=2r_1r_2/(r_1+r_2)$$
Question 10
1 / -0

Taking the gravitational potential at a point infinte distance away as zero, the gravitational potential at a point $$A$$ is $$-5$$ unit. If the gravitational potential at a point infinite distance away is taken as $$+10$$ units, the potential at a point $$A$$ is

A
$$\displaystyle -5$$ unit

B
$$\displaystyle +5$$ unit

C
$$\displaystyle +10$$ unit

D
$$\displaystyle +15$$ unit

Solution

The gravitational potential $$V$$ at a point distance $$'r'$$ from a body a mass $$m$$ is equal to the amount of work done in moving a unit mass from infinity to that point
$$\displaystyle V_r-V_\infty =-\int_{\infty}^{r} \bar{E}.d \bar{r}=-GM (1/r-1/ \infty)$$
$$\displaystyle = \frac {-GM}{r} \left [ As \bar{E}= \frac {-dV}{dr} \right ]$$

$$(i)$$ In the first case
when $$\displaystyle V_{\infty}=0, V_{r}=\frac {-GM}{r}= -5$$ unit
$$(ii)$$ In the second case $$V_{\infty}=+10$$ unit
$$\displaystyle V_{r}-10=-5$$
or $$\displaystyle V_{r}=+5$$ unit

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Gravitation Test - 34

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Gravitation Test - 34

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SHARING IS CARING

Question 1

$$\dfrac { R }{ 2 } $$

$$\sqrt { 2 } { R }$$

$$\left( \sqrt { 2 } -{ 1 } \right) { R }$$

$$\left( \sqrt { 2 } +{ 1 } \right) { R }$$

Solution

Question 2

$$\displaystyle V=0$$ and $$E=0$$

$$\displaystyle V=0$$ and $$E \neq 0$$

$$\displaystyle V\neq 0$$ and $$E = 0$$

All of the above

Solution

Question 3

$$\displaystyle 7$$ $$km/sec$$

$$\displaystyle 112$$ $$km/sec$$

$$\displaystyle 11.2$$ $$km/sec$$

$$\displaystyle 1.1$$ $$km/sec$$

Solution

Question 4

Radius of the planet

The direction of projection

The mass of the planet

None of these

Solution

Question 5

It is closer to the earth

It revolves round the earth

It gets light from the sun

The escape velocity of gas molecules is lesser than their root mean square velocity

Solution

Question 6

$$\displaystyle W_1 < W_2 > W_3$$

$$\displaystyle W_1 = W_2 = W_3$$

$$\displaystyle W_1 < W_2 < W_3$$

$$\displaystyle W_1 > W_2 > W_3$$

Solution

Question 7

$$\cfrac { 1 }{ 2 } mgR$$

$$2mgR$$

$$\cfrac { 1 }{ 4 } mgR$$

$$mgR$$

Solution

Question 8

Zero

Infinite

Same as on the surface on earth

None of these

Solution

Question 9

$$\displaystyle (r_1+r_2)/4$$

$$\displaystyle (r_1+r_2)/(r_1-r_2)$$

$$\displaystyle 2r_1r_2/(r_1+r_2)$$

$$\displaystyle (r_1+r_2)/3$$

Solution

Question 10

$$\displaystyle -5$$ unit

$$\displaystyle +5$$ unit

$$\displaystyle +10$$ unit

$$\displaystyle +15$$ unit

Solution

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