Gravitation Class 11 Notes Physics Chapter 8 - CBSE

What Are Gravitation ?

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Newton’s Law Of Gravitation

If states that every body in the universe attracts every other body with a force which is directly proportional to the product of their mass and inversely proportional to the square of the distance between them.

$$\text{F} = \text{G}\frac{m_{1}×m_{2}}{r^{2}}$$

Where G is the universal gravitational constant.

• G = 6.67 × 10–11 Nm2 /kg2.
• Dimension formula of G is [M–1L3 T–2].
• Gravitational forces:

(i) are independent of intervening medium

(ii) obeys Newton’s third law of motion

(iii) has spherical symmetry

(iv) are independent of the presence of other bodies.

(v) obeys principles of superposition

(vi) are central forces.

Acceleration Due To Gravity (G)

The acceleration produced in the motion of a body under the effect of gravity is called acceleration due to gravity.

At the surface of the earth,

$$\text{g} =\frac{\text{GM}}{\text{R}^{2}}$$

where M is the mass and R is the radius of the earth. Its unit is ms–2.

• Mass of the earth

$$\text{M} =\frac{gR^{2}}{\text{G}}$$

• Mean density of the earth,

$$\rho =\frac{\text{3g}}{4\pi\text{GR}}$$

Weight Of A Body

It is the gravitational force with which a body is attracted towards the centre of the earth.

$$\vec{w} = \vec{\text{mg}}$$

Weight of a body is a vector quantity. It is measured in newton, kg-wt. etc.

Variation Of Acceleration Due To Gravity

(i) Effect of altitude: A a height h

$$\text{g}_h = g\bigg(1 -\frac{2h}{\text{R}}\bigg),\\\text{when h << R}\\\therefore\space \text{g}_h\lt g$$

(ii) Effect of depth: At a depth d

$$g_{d} = g\bigg(1 -\frac{d}{\text{R}}\bigg)$$

∴ gd < g

Gravitational Field

It is the space around a material body in which its gravitational pull can be experienced by other bodies.

Gravitational Potential

The gravitational potential at a point in the gravitational field of a body is the amount of work done in bringing a body of unit mass from infinity to that point. It is a scalar quantity.

$$\text{Graviational potential,}\\\text{V} =\frac{-\text{GM}}{r}$$

S.I. unit of V is Jkg–1.

Gravitational P.E.

It is defined as the energy associated with a body due to its position in the gravitational field of another body and is measured by the amount of work done in bringing a body from infinity to a given point in the gravitational field of the other body.

$$\text{Gravitational P.E. = }\frac{-\text{GM}}{\text{r}}$$

Escape Velocity

It is the minimum velocity with which a body must be projected vertically upward in order that it may just escape the gravitational field of the earth.

$$v_{c} =\sqrt{\frac{\text{2 GM}}{\text{R}}}\\=\sqrt{\text{2gR}}$$

For earth, the value of escape velocity is 11.2 km s–1.

Satellite

It is a heavenly or an artificial body which is revolving continuously in an orbit around a planet.

Orbital Velocity Of A Satellite (V0)

It is the velocity required to put a satellite in its orbit around a planet. The orbital velocity of a satellite revolving around the earth at a height h is given by

$$v_{0} =\sqrt{\frac{\text{GM}}{\text{R + H}}} =\sqrt{\frac{\text{gR}^{2}}{\text{R + H}}}\\=\text{R}\sqrt{\frac{g}{\text{R + h}}}$$

At the close to the surface of the earth (i.e., R>>h)

$$v_{0} =\sqrt{\text{gR}}$$

Time Period Of A Satellite (T)

$$\text{T} =\sqrt{\frac{3\pi(R + h)^{3}}{\text{G}\rho\text{R}^{3}}}$$

If the satellite revolves just close to the surface of the earth h = 0,then

$$\text{T} =\sqrt{\frac{3\pi}{\text{G}\rho}}$$

Height Of A Satellite Above The Earth’s Surface

$$h =\bigg[\frac{\text{T}^{2}\text{R}^{2}}{4\pi^{2}}\bigg]^{\frac{1}{3}}-\text{R}$$

Total energy of a satellite:

$$\text{P.E.} = \text{U} =\frac{-\text{GMm}}{\text{r}}\\\text{K.E} =\frac{1}{2}m\frac{\text{GM}}{r}\\\text{Total energy,}\\\text{E = K + U}\\=\bigg(\frac{-\text{GMm}}{\text{2r}}\bigg)$$

Kepler’s Law

• Law of orbits: Every planet moves in an elliptical orbit around the sun, with the sun being at one of the focii.
• Law of areas: The radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time i.e., the areal velocity of a planet is constant.
• Law of periods: The square of the period of revolution (T) of a planet around the sun is proportional to the cube of the semi-major axis r of the ellipse.
T2 ∝ r3