How Do Keplers Laws Support Newtons Theory of Gravitation Physics Chapter 7 Review

In astronomy, Kepler'south laws of planetary motion are three scientific laws describing the movement of planets around the sun.

• Kepler showtime law – The police force of orbits
• Kepler'due south 2nd police force – The police of equal areas
• Kepler's third police – The police of periods

Table of Content:

• Introduction
• Commencement law
• Second Constabulary
• Third Police

Introduction to Kepler'south Laws

Motion is always relative. Based on the energy of the particle nether move, the motions are classified into two types:

• Divisional Motion
• Unbounded Move

In bounded motion, the particle has negative total energy (Due east<0) and has two or more extreme points where the full energy is ever equal to the potential energy of the particle i.east the kinetic free energy of the particle becomes aught.

For eccentricity 0≤ due east <1, East<0 implies the trunk has bounded motion. A circular orbit has eccentricity east = 0 and elliptical orbit has eccentricity e < i.

In unbounded motion, the particle has positive total free energy (E>0) and has a single extreme point where the total energy is e'er equal to the potential energy of the particle i.eastward the kinetic energy of the particle becomes zero.

For eccentricity east ≥ 1, E > 0 implies the body has unbounded movement. Parabolic orbit has eccentricity eastward = 1 and Hyperbolic path has eccentricity e>1.

⇒ Likewise Read:

• Gravitational Potential Energy
• Gravitational Field Intensity

Kepler's laws of planetary motion can be stated as follows:

Kepler Outset law – The Law of Orbits

According to Kepler'due south first police," All the planets revolve around the sun in elliptical orbits having the sun at 1 of the foci". The point at which the planet is close to the sunday is known as perihelion and the indicate at which the planet is farther from the sun is known every bit aphelion.

It is the characteristics of an ellipse that the sum of the distances of whatsoever planet from two foci is constant. The elliptical orbit of a planet is responsible for the occurrence of seasons.

Kepler First Constabulary – The Law of Orbits

Kepler'south Second Law – The Law of Equal Areas

Kepler's second constabulary states " The radius vector fatigued from the sun to the planet sweeps out equal areas in equal intervals of time"

As the orbit is not round, the planet's kinetic energy is not constant in its path. Information technology has more kinetic free energy virtually perihelion and less kinetic energy nigh aphelion implies more speed at perihelion and less speed (v_min) at aphelion. If r is the distance of planet from sun, at perihelion (r_min) and at aphelion (r_max), then,

r_min+ r_max = 2a × (length of major axis of an ellipse) . . . . . . . (1)

Kepler's 2nd Police – The law of Equal Areas

Using the law of conservation of angular momentum the law can be verified.  At any betoken of time, the angular momentum can be given as, L = mr^twoω.

Now consider a small area ΔA described in a pocket-size time interval Δt and the covered angle is Δθ. Let the radius of curvature of the path be r, then the length of the arc covered = r Δθ.

ΔA = 1/2[r.(r.Δθ)]= 1/2r²Δθ

Therefore, ΔA/Δt = [ i/2r²]Δθ/dt

\(\brainstorm{array}{l}\lim_{\Delta t\rightarrow 0}\frac{\Delta A}{\Delta t}=\frac{ane}{2}r^{2}\frac{\Delta \theta }{\Delta t}\end{array} \)

, taking limits both side as, Δt→0

⇒

\(\begin{array}{l}\frac{dA}{dt}=\frac{one}{two}r^{2}\omega\end{array} \)

\(\brainstorm{array}{50}\frac{dA}{dt}=\frac{L}{2m}\end{array} \)

At present, past conservation of athwart momentum, 50 is a constant

Thus, dA/dt = constant

The area swept in equal interval of time is a constant.

Kepler's 2nd law can also be stated as "The areal velocity of a planet revolving around the dominicus in elliptical orbit remains constant which implies the angular momentum of a planet remains abiding". As the athwart momentum is constant all planetary motions are planar motions, which is a directly upshot of central forcefulness.

⇒ Check: Acceleration due to Gravity

Kepler's Third Police force – The Constabulary of Periods

Co-ordinate to Kepler's law of periods," The square of the time period of revolution of a planet around the sun in an elliptical orbit is straight proportional to the cube of its semi-major axis".

Tⁱⁱ ∝ a^three

Shorter the orbit of the planet around the sun, shorter the time taken to complete one revolution. Using the equations of Newton's law of gravitation and laws of motility, Kepler'southward third law takes a more general form:

P^two= 4π² /[Yard(M_i+ M₂)] × aⁱⁱⁱ

where M₁ and M_ii are the masses of the two orbiting objects in solar masses.

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