Gravitation Test

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Gravitation Test - 18

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Weekly Quiz Competition

Question 1
1 / -0

Planets rotate around the Sun in a path best described as

A
elliptical

B
circular

C
parabola

D
none of the above

Solution

Using Kepler's first law the orbit of every planet is an ellipse with the Sun at one of the two foci.
Question 2
1 / -0

At what height, the value of '$$ g $$' is half that on the surface of the earth of radius $$R$$?

A
$$R$$

B
$$2R$$

C
$$0.414R$$

D
$$0.75R$$

Solution

We have, $$g=\dfrac {GM}{r^2}$$; $$r$$ is the distance from the centre of Earth.
$$\Rightarrow g\propto \dfrac {1}{r^2}$$ above the surface of the earth.
$$\Rightarrow \dfrac {g_s}{g_h}=\dfrac {r_h^2}{r_s^2}$$

Since we want the value of $$g_s$$ to be half, we replace $$g_h$$ by $$g_s/2$$
$$\Rightarrow \dfrac {g_s}{g_s/2}=[\dfrac {R+h}{R}]^2$$
$$\Rightarrow 2=[\dfrac {R+h}{R}]^2$$
$$\Rightarrow \dfrac {R+h}{R}=\sqrt 2$$
$$\Rightarrow \dfrac {h}{R}=1.414-1=0.414$$
$$\Rightarrow h=0.414R$$
Question 3
1 / -0

The difference in $$PE$$ of an object of mass $$10kg$$ when it is taken from a height of $$6400km$$ to $$12800km$$ from the surface of the earth is

A
$$3.11\times 10^{8}$$ J

B
$$1.565\times 10^{8}$$ J

C
$$2.65\times 10^{8}$$ J

D
$$4.5\times 10^{8}$$ J

Solution

Difference in potential energy is given by:
$$PE=-GMm\left(\dfrac {1}{R_1}-\dfrac {1}{R_2}\right)$$
But, $$R_1 = 6400$$ $$km$$ and $$R_2 = 12800$$ $$ km$$
$$\Rightarrow PE=-6.67\times10^{-11}\left(5.97\times10^{24}\right) (10) \left(\dfrac {1}{6400}-\dfrac {1}{12800}\right)\times10^{-3}$$
$$\Rightarrow PE = 0.00311\times10^{11}J$$
Question 4
1 / -0

The kinetic energy needed to project a body of mass $$m$$ from earth's surface $$($$ radius $$R$$ $$)$$ to infinity is

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

B
$$2mgR$$

C
$$mgR$$

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

Solution

For the body to escape Earth's gravitation and reach infinity, the initial kinetic energy has to be just higher than the Gravitational potential energy of the body on the surface.
Total energy of the body just after projecting is:
$$E=\dfrac {1}{2}mv^2 - \dfrac {GMm}{R}
= \dfrac {1}{2}mv^2 - mgR$$
To escape Earth's gravitation, this energy has to be positive:
$$\Rightarrow \dfrac {1}{2}mv^2 - mgR \geq 0
\Rightarrow K.E \geq mgR$$
So, the minimum Kinetic energy required is $$mgR$$.
Question 5
1 / -0

The factor(s) affecting the value of 'g' is(are):

A
depth of the body from earth's surface

B
height of the body from the earth's surface

C
mass of the body

D
both A and B

Solution

The acceleration due to gravity changes with the position of the body with respect to the earth surface. So, 'g' changes with height above the earth's surface and with depth below it as well.
Question 6
1 / -0

A spring balance is on sea level. If a body is weighed with this balance at consecutively increasing heights from earth's surface, the weight indicated by the balance:

A
will go on decreasing continuously

B
will go on increasing continuously

C
will remain same

D
will first increase and then decrease

Solution

As the value of g decreases continuously as we, move consecutively from earth's surface. The weight indicated by a spring balance of a body at a consecutively increasing height from earth's surface also decreases.
Question 7
1 / -0

The minimum velocity of projection to go out from the earth's gravitational pull is called

A
Terminal velocity

B
Escape velocity

C
Angular velocity

D
Orbital velocity

Solution

Escape velocity is the minimum speed needed for an object to "break free" from the gravitational attraction of a massive body(in other words Earth).

The constant maximum velocity reached by a body falling under gravity through a fluid is called terminal velocity.

The minimum velocity required to place or maintain a satellite in a given orbit is called orbital velocity.

The rate of change of angular position of a rotating body. is called angular velocity.
Question 8
1 / -0

A satellite is orbiting around the earth. Then, the plane of the orbit:

A
must be equatorial plane.

B
must pass through the earth's centre.

C
must pass through the poles.

D
any plane with centre lying on the axis of rotation of the earth.

Solution

When ever a body is said to be orbiting about another, it is actually orbiting about the centre of mass of other body. So, in this question for a satellite to orbit around the earth, it must have the earth's centre in its orbiting plane.
Question 9
1 / -0

An object is weighted in the following places using a spring balance. In which place will it weight the heaviest?

A
On the Moon

B
At the equator

C
At the pole

D
In outer space

Solution

The gravitational force at the poles is higher than at the equator. Hence an object appears to be heavier at the poles.
Question 10
1 / -0

At what height in km over the earth's pole, the free fall acceleration decreases by one percent? (Assume the radius of the earth to be 6400 km).

A
32

B
64

C
80

D
1.253

Solution

$$g^1=g/(1+\dfrac {h}{R})^2$$
$$0.99=\dfrac {1}{(1+\dfrac {h}{R})^2}$$
$$0.995=\dfrac {R}{R+h}$$

$$0.995h=0.0050R$$

$$h=32.2 km$$

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Gravitation Test - 18

Result

Gravitation Test - 18

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SHARING IS CARING

Question 1

elliptical

circular

parabola

none of the above

Solution

Question 2

$$R$$

$$2R$$

$$0.414R$$

$$0.75R$$

Solution

Question 3

$$3.11\times 10^{8}$$ J

$$1.565\times 10^{8}$$ J

$$2.65\times 10^{8}$$ J

$$4.5\times 10^{8}$$ J

Solution

Question 4

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

$$2mgR$$

$$mgR$$

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

Solution

Question 5

depth of the body from earth's surface

height of the body from the earth's surface

mass of the body

both A and B

Solution

Question 6

will go on decreasing continuously

will go on increasing continuously

will remain same

will first increase and then decrease

Solution

Question 7

Terminal velocity

Escape velocity

Angular velocity

Orbital velocity

Solution

Question 8

must be equatorial plane.

must pass through the earth's centre.

must pass through the poles.

any plane with centre lying on the axis of rotation of the earth.

Solution

Question 9

On the Moon

At the equator

At the pole

In outer space

Solution

Question 10

32

64

80

1.253

Solution

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