Gravitation Test

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Gravitation Test - 45

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Weekly Quiz Competition

Question 1
1 / -0

The energy required to remove a body of mass $$m$$ from earth's surface is/are equal to:

A
$$\dfrac{-GMm}{R}$$

B
$$mgR$$

C
$$-mgR$$

D
none of these.

Solution

An object can be thrown up with a certain minimum initial velocity so that, the object goes beyond the earth's gravitational field and escape from earth, this velocity escape velocity of the earth.
potential energy at surface is $$-\frac{GMm}{R}$$, so kinetic energy that has to give to reach particle to infinite ( zero energy) is equal to potential energy.
$$E=\frac{GMm}{R}=g.m.R$$
so the best possible answer is option B.
Question 2
1 / -0

When a satellite has an elliptical orbit, the plane of the orbit

A
sometimes passes through the centre of earth

B
does not pass through the centre of earth

C
passes through the centre of earth always

D
none of the above.

Solution

When a satellite has an elliptical orbit, the plane of the orbit always passes through the center of earth.
Question 3
1 / -0

Let $$V$$ and $$E$$ be the gravitational potential and gravitational field at a distance $$r$$ from the centre of a uniform spherical shell. Consider the following two statements, ($$A$$) The plot of $$V$$ against $$r$$ is discontinuous and ($$B$$) The plot of $$E$$ against $$r$$ is discontinuous.

A
Both A and B are correct

B
A is correct but B is wrong

C
B is correct but A is wrong

D
both A and B are wrong.

Solution

Let $$V$$ be the gravitational potential and $$E$$ be the gravitational field at a distance $$r$$ from the center of the sphere.
Graphically as shown in above images.
From IMAGE $$1$$ and IMAGE $$2$$
IMAGE $$1$$ Plot of $$V$$ against $$r$$ is continuous.
IMAGE $$2$$ Plot of $$E$$ against $$r$$ is discontinuous.
Thus condition $$A$$ is wrong and $$B$$ is correct.
Question 4
1 / -0

If a particle is slowly brought from reference point to another point $$P$$ in a gravitational field, then work done per unit mass by the external agent is (at that point)

A
gravitational force

B
gravitational field intensity

C
gravitation potential

D
none of the above

Solution

If a particle is slowly brought from reference point to another point P in a gravitational field, then work done per unit mass by external agent is (at that point) gravitational potential.
Question 5
1 / -0

There is no atmosphere on the moon because

A
it is closer to the earth

B
it revolves around the earth

C
it gets light from the sun

D
the escape velocity of gas molecules is less than their root mean square velocity here

Solution

On moon, due to the lower $$g$$, the escape velocity of gas molecule is less than their root mean square velocity. Thus moon has no atmosphere.
Question 6
1 / -0

The value of g at a particular point is 9.8 $$m/sec^2$$ suppose the earth suddenly shrink uniformly to half its present size without losing any mass. The value of $$g$$ at the same point (assuming that the distance of the point from the centre of the earth does not shrink) will become

A
$$9.8m/sec^2$$

B
$$4.9m/sec^2$$

C
$$19.6m/sec^2$$

D
$$3.1m/sec^2$$

Solution

$$g=\cfrac { GM }{ r } $$
Here $$M$$ and $$r$$ are constant
$$\therefore { g }^{ \prime }=9.8m{ s }^{ -2 }$$
Question 7
1 / -0

A body of mass 'm' is approaching towards the centre of a hypothetical hollow planet of mass 'M' and radius 'R'. The speed of the body when it passes the centre of the planet through a diametrical tunnel is:

A
$$\sqrt {\dfrac{GM}{R}}$$

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

C
Zero

D
none of these.

Solution
Question 8
1 / -0

If g is the acceleration due to gravity at the Earths, surface, the gain of the potential energy of an object of mass m raised from the surface of the Earth to height equal to the radius R of the Earth is :

A
$$\dfrac{mgR}{4}$$

B
$$\dfrac{mgR}{2}$$

C
$$mgR$$

D
$$2 mgR$$

Solution

Potential energy of an object at the surface of the Earth is
$$U_1=-\dfrac{GMm}{R}$$ ...(i)
Potential energy of the object at a height, h = R from the surface of the Earth.
$$U_2=-\dfrac{GMm}{R+h}=-\dfrac{GMm}{R+R}$$
Gain in potential energy
$$\triangle U=U_2-U_1=-\dfrac{GMm}{2R}+\dfrac{GMm}{R}=\dfrac{1}{2}\dfrac{GMm}{R}$$
Also, $$GM=gR^2$$
$$\triangle U=\dfrac{1}{2}\dfrac{gR^2m}{R}=\dfrac{mgR}{2}$$
Question 9
1 / -0

The ratio of the acceleration due to gravity as the bottom of a deep mine and that on the surface of the earth is 978/980. Find the depth of the mine, if the density of the earth is uniform throughout and the radius of the earth is 6300 km.

A
12.88 km

B
13.0 km

C
25.38 km

D
90.9 km

Solution

Ratio of acceleration due to gravity
$$ \dfrac { g'}{g} = \dfrac {978}{9800} = I - \dfrac {d}{R} $$
$$ \dfrac {d}{R} = I - \dfrac {978}{9800} $$
$$ d = \dfrac {2}{980} R $$
$$ = \dfrac {2 \times 6300}{980} $$
= 12.86 km
Question 10
1 / -0

The escape velocity from the Earth is $$11 kms^{-1}$$. The escape velocity from a planet having twice the radius and same mean density as that of Earth is :

A
$$11 kms^{-1}$$

B
$$5.5 kms^{-1}$$

C
$$22 kms^{-1}$$

D
$$10 kms^{-1}$$

Solution

Escape velocity
$$v_e=\sqrt{\dfrac{2GM}{R}}=R\sqrt{\dfrac{8}{3}\pi Gp}$$
$$\therefore v_e \propto R$$ if p = constant
Since, planet has double the radius in comparison to the Earth
$$\therefore v_e=22 kms^{-1}$$

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Gravitation Test - 45

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Gravitation Test - 45

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SHARING IS CARING

Question 1

$$\dfrac{-GMm}{R}$$

$$mgR$$

$$-mgR$$

none of these.

Solution

Question 2

sometimes passes through the centre of earth

does not pass through the centre of earth

passes through the centre of earth always

none of the above.

Solution

Question 3

Both A and B are correct

A is correct but B is wrong

B is correct but A is wrong

both A and B are wrong.

Solution

Question 4

gravitational force

gravitational field intensity

gravitation potential

none of the above

Solution

Question 5

it is closer to the earth

it revolves around the earth

it gets light from the sun

the escape velocity of gas molecules is less than their root mean square velocity here

Solution

Question 6

$$9.8m/sec^2$$

$$4.9m/sec^2$$

$$19.6m/sec^2$$

$$3.1m/sec^2$$

Solution

Question 7

$$\sqrt {\dfrac{GM}{R}}$$

$$\sqrt {\dfrac{2GM}{R}}$$

Zero

none of these.

Solution

Question 8

$$\dfrac{mgR}{4}$$

$$\dfrac{mgR}{2}$$

$$mgR$$

$$2 mgR$$

Solution

Question 9

12.88 km

13.0 km

25.38 km

90.9 km

Solution

Question 10

$$11 kms^{-1}$$

$$5.5 kms^{-1}$$

$$22 kms^{-1}$$

$$10 kms^{-1}$$

Solution

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