Physics : Gravitation : Book Back Important Questions, Answers, Solutions : Long Questions and Answer

__Gravitation (Physics)__

__Long
Answer Questions__

** **

**1.
Discuss the important features of the law of gravitation.**

**Important
features of gravitational force:**

●
As the distance between two masses increases, the strength of the force tends
to decrease because of inverse depends on r^{2}.

●
Uranus experiences less gravitational force from the Sun than the Earth since
Uranus is at larger distance from the Sun compared to Earth.

●
The gravitational forces between two particles always constitute an action
reaction pair.

●
The gravitational force exerted by the Sun on the Earth is always towards the
Sun.

●
The reaction- force is exerted by the sun on the Earth on the Sun.

●
The direction of this reaction force is towards Earth.

● The torque experienced by angular momentum is a constant vector.

●
The angular momentum of the Earth about the Sun is constant throughout the
motion

●
Earth orbits around the Sun due to Sun's gravitational force,

●
To calculate force of attraction between hollow sphere of mass M with uniform
density and point mass m kept outside the hollow sphere, we can replace the
hollow sphere of mass M as equivalent to a point mass M located at the centre
of the hollow sphere.

●
If we place another object of mass 'm' inside this hollow sphere, the force
experiencedby this mass 'm' will be zero.

** **

**2.
Explain how Newton arrived at his law of gravitation from Kepler’s third law.**

**Newton’s inverse square Law:**

Newton
considered the orbits of the planets as circular. For circular orbit of radius
r, the centripetal acceleration towards the centre is

a
= - v^{2} / r ……………(1)

Here
v is the velocity and r, the distance of the planet from the centre of the
orbit. The velocity in terms of known quantities r and T, is

v
= 2πr / T ………… (2)

Here
T is the time period of revolution of the planet. Substituting this value of v
in equation (1) we get.

a
= (2πr / T)^{2} / r = − 4π^{2}r / T^{2} …….. (3)

Substituting
the value of ‘a’ from (3) in Newton's second law, F=ma, where ‘m' is the mass
of the planet.

F=
− 4π^{2}mr / T^{2
}………….(4)

From
Kepler's third law

r^{3}/
T^{2} = K (constant) ……….(5)

r
/ T^{2 }= K /r^{2 }……….(6)^{}

By
substituting equation 6 in the force expression, we can arrive at the law of
gravitation.

F=
− 4π^{2}mK / r^{2} ……….(7)

●
Negative sign implies that the force is attractive and it acts towards the
centre.

●
According to Newtons third law, if Earth is attracted by the Sun, then the Sun
must also be attracted by the Earth with the same magnitude of force.

●
The Sun's mass (M) should also occur explicitly in the expression for force.

●
From this insight, he equated the constant to GM which turned out to be law of
gravitation.

F
= − GMm / r^{2}

The
negative sign implies that the gravitational force is attractive.

** **

**3.
Explain how Newton verified his law of gravitation.**

●
Newton verified his law of universal gravitation by comparing the acceleration
of a terrestrial object to the acceleration of the moon.

●
He knew that the distance from the centre of earth to the centre of two spheres
of known mass at either end of a light rod suspended by a thin fibre from the
centre of the rod.

●
He found earlier that the small force that was needed to twist the fibre.

●
By bringing a third sphere close to one of the suspended spheres.

●
He was able to measure the force of gravity between the spheres and hence
gravitation.

** **

**4.
Derive the expression for gravitational potential energy.**

●
Among the two masses, m_{1} to be fixed in its position, work must be
done on m_{2} to move the distance from r’ to r.

**Two
distant masses changing the linear distance**

●
To move the mass m_{2} through an infinitesimal displacement d from to + d, work has to be done externally. This infinitesimal work
is given by ………(1)

●
The work is done against gravitational force, therefore,

Substituting
Equation (2) in (1), we get

∴ *dW* =* *[ *Gm _{1}m_{2}*

Thus
the total work done for displacing the particle from r' to r is

*W* = ʃ^{r}_{r’}*dW
*= ʃ^{r}_{r’}* *[
*Gm _{1}m_{2}*

*W* = − (*Gm _{1}m_{2}*

*W* = − [ *Gm _{1}m_{2}*

*W* = *U*(*r*) - *U*(*r'*)

where
*U*(*r*) = [−*Gm _{1}m_{2}*

This
work done W gives the gravitational potential energy difference of the system
of masses m_{1} and m_{2} when the separation between them are
r and r' respectively

**Case
1:**
If r < r'

Since
gravitational force is attractive, m_{2} is attracted by m_{1}.
Then m_{2} can move from r' to r without any external work. Here work
is done by the system spending its internal energy and hence the work done is
said to be negative.

**Case
2:**
If r>r'

●
Work has to be done against gravity to move the object from r' to r. Therefore
work is done on the body by external force and hence work done is positive.

●
Now gravitational potential energy can be discussed by choosing one point as
the reference point.

●
Let us choose r . Then the second term in the equation becomes zero.

*W* = − [ *Gm _{1}m_{2}*

●
Now we can define gravitational potential energy of a system of two masses m_{1}
and m_{2} separated by a distance r as the amount of work done to take
the mass m_{2} from a distance r to infinity assuming m_{1} to
be fixed in its position and is written as *U *(*r*) = - [ *Gm _{1}m_{2}*

●
Here we choose U(∝) = 0 as
the reference point.

●
The gravitational potential energy U(r) is always negative.

● The unit of gravitational potential energy U (r) is Joule and it is a scalar quantity.

** **

**5.
Prove that at points near the surface of the Earth, the gravitational potential
energy of the object is U = mgh**

●
Consider the Earth and mass system, with r, the distance between the mass m and
the Earth's centre. Then the gravitational potential energy.

h<<R

U
= − [ GM_{e}m / r ] ……….(1)

●
r = R_{e} + h R_{e} is the radius of the Earth. h is the height
above the Earth's surface,

U
= − G [ M_{e}m / (R_{e}+h) ] ……….(2)

If
h << Re, equation (2) can be modified as

U
= -G [ M_{e}m / R_{e}( 1+h / R )_{e} ]

U
= − G [ M_{e}m / R_{e} ] ( 1+h / R_{e })^{−1} …………….(3)

●
Using Binomial expansion and neglecting the higher order terms, we get

U
= − G [ M_{e}m / R_{e} ] ( 1- h/R_{e}) …………….(4)

We
know that, for a mass m on the Earth's surface

G
[ M_{e}m / R_{e} ] = mgR_{e }…………….(5)

Substituting
equation (5) in (4) we get,

U
= − mgR_{e} + mgh …………(6)

The
first term in the above expression is independent of the height h. For example,
if the object is taken from height h_{1 }to h_{2}, then the
potential energy at h_{1} is

U(h_{1})
= −mgR_{e }+ mgh_{1} ………… (7)

and
the potential energy at h_{2} is

U(h_{2})
= −mgR_{e }+ mgh_{2} ………… (8)

The
potential energy difference between h_{l} (doubt)and h_{2 }is

U(h_{2})
− U(h_{1}) = mg (h_{2} − h_{1}) . ………….. (9)

Omitting
the first term (1) and (2) we get.

U
= mgh

** **

**6.
Explain in detail the idea of weightlessness using lift as an example.**

●
When the lift falls (when the lift wire cuts) with downward acceleration a=g,
then person inside the elevator is in state of weightlessness or free fall.

●
As they fall freely, they are not in contact with any surface (by neglecting air
friction).

●
The normal force acting on the object is zero.

●
The downward acceleration is equal to the acceleration due to the gravity of
the Earth.

**i.e** (a=g).
From equation N=m (g-a) we get,

a
= g

∴ N = m
(g-g) = 0.

This
is called the state of weightlessness.

** **

**7.
Derive an expression for escape speed.**

●
Consider an object of mass M on the surface of the Earth. When it is thrown up
with an initial speed v_{i}, the initial total energy of the object is

E_{i}
= 1/2 MV_{i}^{ 2} – [ GMM_{E }/ R_{E} ] ………. (1)

Where
M_{E}, is the mass of the Earth and R_{E} -the radius of the
Earth.

The
term −GMM_{E }/ R_{E} is the potential energy of the mass M.

●
When the object reaches a height far away from Earth and hence treated as
approaching infinity, the gravitational potential energy becomes zero [U(∞)=0]
and the kinetic energy becomes zero as well.

●
Therefore the final total energy of the object becomes zero.

●
This is for minimum energy and for minimum speed to escape. Otherwise kinetic
energy can be non- zero.

E_{f
}= 0, According to the law of energy conservation,

E_{i}
= E_{f }…………. (2)

Substituting
(1) in (2) we get.

1/2
*MV _{i}*

1/2
*MV _{i}*

●
The escape speed, the minimum speed required by an object to escape from
Earth's gravitational field, replacing,

*v*_{i }with v_{e}.
i.e

1/2
*MV _{e}*

*V _{e}*

*V _{e}*

Using
*g *= *GM _{E }/ R*

*V _{e}*

*V _{e}*

From
equation (5) the escape speed depends on two factors: acceleration due to
gravity and radius of the Earth. It is completely independent of the mass of
the object.

** **

**8.
Explain the variation of g with lattitude.**

**Variation
of g with latitude:**

●
Whenever we analyze the motion of objects in rotating frames, we must taken
into account the centrifugal force.

●
Even though the Earth is an inertial frame, it is not exactly correct because
the Earth spins about its own axis.

●
So when an object is on the surface of the Earth, it experiences a centrifugal
force that depends on the latitude of the object on Earth. If the Earth were
not spinning, the force on the object would have been mg. However, the object
experiences an additional centrifugal force due to spinning of the Earth.

●
This centrifugal force is given by mω^{R}R'

R'
= R cos λ ……………. (1)

Where
λ is the latitude. The component of centrifugal acceleration experienced by the
object in the direction opposite to g is

a_{c}
= ω^{2}R' cos λ = ω^{2}*R* cos^{2} λ since R' cos λ

Therefore,
g' = g − ω^{2}*R* cos^{2} λ ………. (2)

From
the expression (2). At equator, λ = 0; g' = g − ω^{2}R. The
acceleration due to gravity is minimum. At poles g' = g, it is maximum. At the
equator, g' minimum.

** **

**9.
Explain the variation of g with altitude.**

**Variation
of g with altitude:**

Consider
an object of mass m at a height h from the surface of the Earth.

Acceleration
experienced by the object due to Earth is

*g'* = GM / (R_{e}
+ h)^{2}

If
h << R_{e} Using Binomial expansion, taking the terms upto first
order

*g'* < g.
As altitude h increase, the acceleration due to gravity g decreases.

** **

**10.
Explain the variation of g with depth from the Earth’s surface.**

**Variation
of g with depth:**

Consider
a particle of mass m which is in a deep mine on the earth. Ex. Coal mines -
Neyveli). Assume the depth of the mine as d. The part of the Earth which is
above the radius (R_{e}-d) do not contribute to the acceleration.

* g*' = *GM*'/ (*R _{e}
*–

*p
*=
M' / V' ; M' / V' = M / V and M' = [M / V] V'

g'
< g As depth increases, decreases.

** **

**11.
Derive the time period of satellite orbiting the Earth.**

**Time
period of the satellite:**

●
The distance covered by the satellite during one rotation in its orbit is equal
to 2π(R_{E}+h) and time taken for it is the time period. T. Then

*v* = *Distance
travelled* / *Time taken* = 2π (*R _{E}+h*) /

From
equation,

Squaring
both side, of the equation (2), we get
*T*^{2}= 4*π ^{2}* /

4*π ^{2}*
/

●
Equation (3) implies that a satellite orbiting the Earth has the same relation
between time and distance as that of Kepler's law of planetary motion. For a
satellite orbiting near the surface of the Earth, h is negligible compared to
the radius of the Earth R_{E}.

*T* = 2* π
√*[*R _{E }*/

** **

**12.
Derive an expression for energy of satellite.**

●
The total energy of a satellite orbiting the Earth at a distance h from the
surface of Earth

●
The total energy of the satellite is the sum of its kinetic energy and the
gravitational potential energy.

●
The potential energy of the satellite is,

*U
*=
*GM _{S}M_{E}*

M_{S}
- mass of the satellite,

M_{E}
- mass of the Earth,

R_{E}
-radius of the Earth.

●
The kinetic energy of the satellite is

KE
= 1/2 M_{S} *v*^{2} ………. (2)

*v* is the
orbital speed of the satellite is equal to *v* = √* *[ *GM _{E}*

●
Substituting the value of v in (2), the kinetic energy of the satellite
becomes,

●
Therefore the total energy of the satellite is

●
The negative sign in the total energy implies that the satellite is bound to
the Earth and it cannot escape from the Earth.

** **

**13.
Explain in detail the geostationary and polar satellites.**

**Geo-stationary
and polar satellite**

●
The satellites orbiting the Earth have different time periods corresponding to
different orbital radii.

●
Kepler's third law is used to find the radius of the orbit.

*T*^{2} = (4π^{2}/*GM _{E}*

(*R _{E
}*+

*R _{E
}*+

●
Substituting for the time period (24 hrs= 86400 seconds), mass and radius of
the Earth, h turns out to be 36,000 km. Such satellites are called **"geo-stationary
satellites",** since they appear to be stationary when seen from Earth.

●
Geo-stationary satellite for the purpose of telecommunication. Another type of
satellite which is placed at distance of 500 to 800 km from the surface of the
Earth orbits the Earth from north to south direction.

●
This type of satellite use that orbits Earth from North pole to South pole is
called a polar satellite. The time period of a polar satellite is nearly 100
minutes and the satellite completes many revolutions in a day.

●
A Polar satellite covers a small strip of area pole to pole during one
revolution. In the next revolution it covers a different strip of area since
the Earth would have moved by a small angle.

** **

**14.
Explain how geocentric theory is replaced by heliocentric theory using the idea
of retrograde motion of planets.**** **

●
Ptolemy introduced the concept of "epicycle" in his geocentric model.
According to this theory, while the planet orbited the Earth, it also underwent
another circular motion termed as "epicycle".

●
A combination of epicycle and circular motion around the Earth gave rise to
retrograde motion of the planets with respect to Earth.

●
But ptolemy's model became more and more complex as every planet was found to
undergo retrograde motion.

●
According to Copernicus model, the Sun is at the centre of the solar system and
all planets orbited the Sun.

●
The retrograde motion of planets with respect to Earth is because of the
relative motion of the planet with respect to Earth.

**15.
Explain in detail the Eratosthenes method of finding the radius of Earth.**

●
During noon time of summer solstice the Sun's rays cast no shadow in the city
Syne which was located 500 miles away from Alexandria.

●
At the same day and same time in Alexandria, the Sun's rays made 7.2 degree
with local vertical

●
This difference of 7.2 degree was due to the curvature of the Earth.

●
The angle 7.2 degree is equivalent to 1/8 radian. So, θ = 1/8 rad.

●
If S is the length of the arc between the cities of Syne and Alexandria, and if
R is radius, then , *S*=*Rθ*=500 miles, so radius of the Earth

●
*R* = 500 *miles* / (1/8)

R
= 4,000 miles

1
mile is equal to 1.609 km. The radius of the Earth to be equal to *R*=6436*km*,
which is amazingly close to the correct value of 6378 km.

**16.
Describe the measurement of Earth’s shadow (umbra) radius during total lunar
eclipse**

●
It is possible to measure the radius of shadow of the Earth at the point where
the Moon crosses.

●
When the Moon is inside the umbra shadow, it appears red in color. As soon as
the Moon exist from the umbra shadow, it appears in crescent shape.

●
By finding the apparent radii of the Earth's umbra shadow and the Moon, the
ratio of the these radii can be calculated.

●
The apparent radius of Earth's umbra shadow = *R _{S}* = 13.2 cm

●
The apparent radius of the moon R_{m} = 5.15 cm

The
ratio *R _{s} / R_{m}* ≈ 2.56.

The
radius of the Earth's umbra shadow is R_{s} = 2.56 × R_{m}

●
The radius of Moon, R_{m} = 1737 km

●
The radius of the Earth's umbra shadow is R_{s} = 2.56 × 1737 km ≡ 4446
km

●
The correct radius is 4610 km

●
Percentage error = [ (4610 – 4446) / 4610 ] × 100 = 35 %

The
error will be reduced if the pictures are taken using a high quality telescope.

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