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The direction of the gravitational intensity at an arbitrary point P for hemispherical shell of uniform mass density is indicated by the arrow:

(i) d
(ii) e
(iii) f
(iv) g


Gravitational potential (V) is constant at all points in a spherical shell. Hence, the gravitational potential gradient (dV/dR) is zero everywhere inside the spherical shell. The gravitational potential gradient is equal to the negative of gravitational intensity. Hence, intensity is also zero at all points inside the spherical shell. This indicates that gravitational forces acting at a point in a spherical shell are symmetric. 

If the upper half of a spherical shell is cut out (as shown in the given figure), then the net gravitational force acting on a particle at an arbitrary point P will be in the downward direction.



Since gravitational intensity at a point is defined as the gravitational force per unit mass at that point, it will also act in the downward direction. Thus, the gravitational intensity at an arbitrary point P of the hemispherical shell has the direction as indicated by arrow e.

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Choose the correct answer from among the given ones:The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 8.12) (i) a, (ii) b, (iii) c, (iv) O.


Gravitational potential (V) is constant at all points in a spherical shell. Hence, the gravitational potential gradient (dV/dR) is zero everywhere inside the spherical shell.

The gravitational potential gradient is equal to the negative of gravitational intensity. Hence, the intensity is also zero at all points inside the spherical shell. This indicates that gravitational forces acting at a point in a spherical shell are symmetric. 

If the upper half of a spherical shell is cut out (as shown in the given figure), then the net gravitational force acting on a particle located at centre O will be in the downward direction.

             

Since gravitational intensity at a point is defined as the gravitational force per unit mass at that point, it will also act in the downward direction.

Thus, the gravitational intensity at centre O of the given hemispherical shell has the direction as indicated by arrow 
c.


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How will you ‘weigh the sun’, that is estimate its mass? The mean orbital radius of the earth around the sun is 1.5 × 108 km.

Orbital radius of the Earth around the Sun, r = 1.5 × 1011 m

Time taken by the Earth to complete one revolution around the Sun, 

T = 1 year

  = 365.25 days 

  = 365.25 × 24 × 60 × 60 s 

Universal gravitational constant, G = 6.67 × 10–11 Nm2 kg–2 

Thus, mass of the Sun can be calculated using the relation, 

M =
   =  

    = 2 × 1030 kg 

Hence, the mass of the Sun is 2 × 1030 kg.
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A Saturn year is 29.5 times the earth year. How far is the Saturn from the sun if the earth is 1.50 ×108 km away from the sun?


Distance of the Earth from the Sun, re = 1.5 × 108 km 
                                                           = 1.5 × 10
11 m  

Time period of the Earth = 

Time period of Saturn, Ts = 29. 5 T

Distance of Saturn from the Sun = r

From Kepler’s third law of planetary motion, 
 
      T = (4π2r3 / GM)1/2 

For Saturn and Sun, 

rs3 / re3  =  Ts2 / Te

         rs = re(Ts / Te)2/3 

            = 1.5 × 1011 (29.5 Te / Te)2/3 

             = 1.5 × 1011 (29.5)2/3 

             = 14.32 × 1011 m 

Hence, the distance between Saturn and the Sun is 1.43 × 1012 m.
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Which of the following symptoms is likely to afflict an astronaut in space
(a) swollen feet,
(b) swollen face,
(c) headache,
(d) orientational problem?

(a) When a person is in standing position, legs hold the entire mass of a body due to gravitational pull. While in space, an astronaut feels weightlessness because of the absence of gravity. Therefore, swollen feet of an astronaut do not affect him/her in space.

(b) The apparent weightlessness in space causes swollen face.

Sense organs such as eyes, ears nose, and mouth constitute a person’s face. This symptom can affect an astronaut in space.


(c) Headaches are caused because of mental strain. It can affect the working of an astronaut in space.

(d) Space has different orientations. Therefore, the orientational problem can affect an astronaut in space.
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