Gravitation Test

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Gravitation Test - 35

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Question 1
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The mass of the moon is $$1/81$$ of earth's mass and its radius $$1/4$$ that of the earth. If the escape velocity from the earth's surface is $$11.2$$ km/sec. its value from the surface of the moon will be

A
$$\displaystyle 0.14 kms^{-1}$$

B
$$\displaystyle 0.5 kms^{-1}$$

C
$$\displaystyle 2.5 kms^{-1}$$

D
$$\displaystyle 5.0 kms^{-1}$$

Solution

$$\displaystyle v_e = \sqrt {\frac{2GM_e}{R_e}}$$
$$v_m = \sqrt {\dfrac{2G \dfrac{M_e}{81}}{\dfrac {R_e}{4}}}= \dfrac {2}{9}v_e$$
$$\displaystyle =2/9 \times 11.2 kms^{-1}=2.5 kms^{-1}$$
Question 2
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The gravitational potential difference between the surface of a planet and a point $$20 m$$ above the surface is $$2$$ joule/kgs. If the gravitational field is uniform, then the work done in carrying a $$5$$ kg body to a height of $$4 m$$ above the surface is

A
$$\displaystyle 2 J$$

B
$$\displaystyle 20 J$$

C
$$\displaystyle 40 J$$

D
$$\displaystyle 10 J$$

Solution

For uniform gravitational field,
$$\displaystyle g =-\dfrac {V}{r}= - \frac {-2}{20}=\frac {1}{10}ms^{-2}$$

Now, $$\displaystyle W= mgh = 5 \times \frac {1}{10}\times 4 = 2J$$
Question 3
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In the region of only gravitational fields of mass '$$M$$' a particle is shifted from $$A$$ to $$B$$ via three different paths of length 5m, 10m and 25m. The work done in different paths is $$W_1, W_2, W_3$$ respectively 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

The gravitational field is a conservative field i.e. work done by gravitational field to displace a body is independent of its path. In other words, work done is determined by difference in potential energy at initial and final positions of body.
Therefore, $$ W = P_f - P_i $$.($$ P_f $$ =final potential energy, $$ P_i $$ = Initial potential energy )
So, $$ W_1 = P_B - P_A $$ and similarly
$$ W_2 = P_B - P_A $$
$$ W_3 = P_B - P_A $$
Thus $$ W_1 = W_2 = W_3 $$
Question 4
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The radius of a planet is $$1/4^{th}$$ of $$R_e$$ and its acc, due to gravity is $$2g$$. What would be the value of escape velocity on the planet, if escape velocity on earth is $$v_e$$

A
$$\displaystyle \frac {v_e}{\sqrt {2}}$$

B
$$\displaystyle v_e \sqrt {2}$$

C
$$\displaystyle 2v_e$$

D
$$\displaystyle \frac {v_e}{2}$$

Solution

The escape velocity on the earth is defined as
$$\displaystyle v_e = \sqrt {2g_eR_e}$$
Where $$R_e$$ & $$g_e$$ are the radius & acceleration due to gravity of earth
Now for planet $$\displaystyle g_p=2g_e, R_p=R_e/4$$
So $$\displaystyle v_p =\sqrt {2g_pR_p}=\sqrt {2 \times 2g_e \times R_e/4}=\frac {v_e}{\sqrt {2}}$$
Question 5
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A satellite of mass '$$m$$', moving around the earth in a circular orbit of radius $$R$$, has angular momentum $$L$$. The areal velocity of satellite is :

A
$$\displaystyle L/2m$$

B
$$\displaystyle L/m$$

C
$$\displaystyle 2L/m$$

D
$$\displaystyle 2L/m_e$$

Solution

According to kepler's second law, areal velocity is constant .
i.e. $$ A_v = \dfrac{rv}{2} = \dfrac{\omega r^2}{2} = \dfrac{L}{2m}$$
Question 6
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Consider earth to be a homogeneous sphere. Scientist $$A$$ goes deep down in a mine and scientist $$B$$ goes high up in a balloon. The gravitational field measured by

A
$$A$$ goes on decreasing and that by $$B$$ goes on increasing

B
$$B$$ goes on decreasing and that by $$A$$ goes on increasing

C
Each decreases at the same rate

D
Each decreases at different

Solution

$$Consider\quad Earth\quad to\quad be\quad a\quad uniform\quad solid\quad sphere.\\ Inside\quad the\quad sphere\quad g\quad \propto \quad R,\quad while\\ outside\quad the\quad sphere\quad g\propto \quad \dfrac { 1 }{ { R }^{ 2 } } \\ Hence,\quad the\quad observatvion\quad of\quad each\quad scientist\quad will\quad differ.$$
Question 7
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The amount of work done in lifting a mass '$$m$$' from the surface of the earth to height $$2R$$ is

A
$$\displaystyle 2mgR$$

B
$$\displaystyle 3mgR$$

C
$$\displaystyle \frac {3}{2}mgR$$

D
$$\displaystyle \frac {2}{3}mgR$$

Solution

Work done $$\displaystyle = U_f-U_i$$

$$\displaystyle =-\dfrac {GMm}{2R+R}-\left ( -\dfrac {Gmm}{R} \right )=\dfrac {2}{3}\dfrac {GmM}{R}=\dfrac {2}{3}mgR$$
Question 8
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At what height from the surface of earth will the value of g be reduced by $$36$$% from the value at the surface? $$R = 6400km$$.

A
$$400 km$$

B
$$800 km$$

C
$$1600 km$$

D
$$3200 km$$

Solution

$${g}'=0.64g=\displaystyle\dfrac{g}{\left ( 1+\dfrac{h}{R} \right )}^{2}$$

$$\Rightarrow 1+\dfrac{h}{R}=\dfrac{5}{4}$$

$$h=\dfrac{R}{4}=1600\ km$$
Question 9
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The total mechanical energy E possessed by a body of mass 'm', moving with a velocity 'v' at a height 'h' is given by : $$E\, =\, \displaystyle \frac{1}{2}\, mv^{2}\, +\, mgh.$$ Make 'm' the subject of formula.

A
$$m\, =\, \displaystyle \frac{2E}{v^{2}\, +\, gh}$$

B
$$m\, =\, \displaystyle \frac{2E}{v^{2}\, +\, 2gh}$$

C
$$m\, =\, \displaystyle \frac{3E}{v^{2}\, +\, 2gh}$$

D
$$m\, =\, \displaystyle \frac{E}{v^{2}\, +\, 2gh}$$

Solution

$$E= \frac{1}{2}mv^{2}+mgh$$

Multiply both side 2:
$$2E= mv^{2}+2mgh$$
Or $$2E= m\left ( v^{2}+2gh \right )$$
Or $$m= \dfrac{2E}{v^{2}+2gh}$$
Question 10
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The change in potential energy, when a body of mass $$m$$ is raised to a height $$nR$$ from the earth's surface is $$(R=$$ radius of earth$$)$$

A
$$\displaystyle mgR \left ( \frac {n}{n-1}\right )$$

B
$$\displaystyle nmgR$$

C
$$\displaystyle mgR \left ( \frac {n^2}{n^2+1}\right )$$

D
$$\displaystyle mgR \left ( \frac {n}{n+1}\right )$$

Solution

$$\displaystyle \bigtriangleup U=U_f-U_i=-\dfrac {GMm}{nR+h}-\left (-\dfrac {GMm}{R}\right )$$

$$\displaystyle =\dfrac {n}{n+1}.\dfrac {GMm}{R}=\dfrac {n}{n+1}mgR$$

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Gravitation Test - 35

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Gravitation Test - 35

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SHARING IS CARING

Question 1

$$\displaystyle 0.14 kms^{-1}$$

$$\displaystyle 0.5 kms^{-1}$$

$$\displaystyle 2.5 kms^{-1}$$

$$\displaystyle 5.0 kms^{-1}$$

Solution

Question 2

$$\displaystyle 2 J$$

$$\displaystyle 20 J$$

$$\displaystyle 40 J$$

$$\displaystyle 10 J$$

Solution

Question 3

$$\displaystyle W_1=W_2=W_3$$

$$\displaystyle W_1>W_2>W_3$$

$$\displaystyle W_1=W_2>W_3$$

$$\displaystyle W_1 < W_2 < W_3$$

Solution

Question 4

$$\displaystyle \frac {v_e}{\sqrt {2}}$$

$$\displaystyle v_e \sqrt {2}$$

$$\displaystyle 2v_e$$

$$\displaystyle \frac {v_e}{2}$$

Solution

Question 5

$$\displaystyle L/2m$$

$$\displaystyle L/m$$

$$\displaystyle 2L/m$$

$$\displaystyle 2L/m_e$$

Solution

Question 6

$$A$$ goes on decreasing and that by $$B$$ goes on increasing

$$B$$ goes on decreasing and that by $$A$$ goes on increasing

Each decreases at the same rate

Each decreases at different

Solution

Question 7

$$\displaystyle 2mgR$$

$$\displaystyle 3mgR$$

$$\displaystyle \frac {3}{2}mgR$$

$$\displaystyle \frac {2}{3}mgR$$

Solution

Question 8

$$400 km$$

$$800 km$$

$$1600 km$$

$$3200 km$$

Solution

Question 9

$$m\, =\, \displaystyle \frac{2E}{v^{2}\, +\, gh}$$

$$m\, =\, \displaystyle \frac{2E}{v^{2}\, +\, 2gh}$$

$$m\, =\, \displaystyle \frac{3E}{v^{2}\, +\, 2gh}$$

$$m\, =\, \displaystyle \frac{E}{v^{2}\, +\, 2gh}$$

Solution

Question 10

$$\displaystyle mgR \left ( \frac {n}{n-1}\right )$$

$$\displaystyle nmgR$$

$$\displaystyle mgR \left ( \frac {n^2}{n^2+1}\right )$$

$$\displaystyle mgR \left ( \frac {n}{n+1}\right )$$

Solution

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