3.2 Kepler's Law

MODUL PDP SPM
PASCA PKP
NEGERI PERAK
2020

Physics Form 4

3.2: KEPLER’S LAWS

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3.2 Kepler’s Laws

Learning standard:

Pupils are able to:

• explain Kepler’s Laws.
• express Kepler’s Third Law, , 2 ∝ 3
• solve problems using Kepler’s Third Law

2

notE

1. Picture shows a German astronomist, mathematician and astrologist: Johannes
Kepler.

Johannes Kepler

2. There are three Kepler’s Laws:
(i) Kepler’s First Law:
All planets move in elliptical orbits with the Sun at one focus
(Law of Orbits).
o The planets in the Solar System have elliptical shaped orbits as shown
in Diagram 1.
o Major axis > minor axis
o Most orbits of the planets have major axis ≈ minor axis (of almost the
same length), thus the shape of the elliptical orbit of the planets is
almost round.
o As such, planets can be assumed to make circular motion around the
Sun.
o Radius of orbit = the average value of the distance between the planet
and the Sun.

Minor axis Shape of the
elliptical orbit

Elliptical focus

Radius of orbit Major
axis
(average
value)

Diagram 1: Orbit of planet around the Sun

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(ii) Kepler’s Second Law:
A line that connects a planet to the Sun sweeps out equal areas in equal
times (Law of Areas).

o tAB = tCD

time taken to move from A to B = time taken to move from C to D
o Area AFB = area CFD
o Distance AB > distance CD

o

Distance Are Area CFD Distance
AB a CD

Higher Lower
linear linear
speed speed

Diagram 2: Motion of planet in its orbit
(iii) Kepler’s Third Law:

The square of the orbital period of any planet is directly proportional to the
cube of the radius of its orbit (Law of Periods).

o ∝
= orbital period of a planet
= radius of orbit

o Larger radius of orbit, longer orbital period.
o Planets which are further from the Sun, time taken to complete one

orbit around the Sun is longer.
o Time taken to around the Sun (1 complete orbit):

(i) Mercury = 0.2 years
(ii) Earth = 1 year (365 days)
(iii) Saturn = 29.5 years
(iv) Neptune = 164.8 years

Diagram 3: Orbits and orbital periods of planets

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3. Newton’s Universal Law of Gravitation + concept of circular motion = Kepler’s Third
Law
o Planets around the Sun = circular motions (as shown in Diagram 4)

Diagram 4: Orbit of a planet
o

Centripetal force = Gravitational force
between the Sun
and the planet

= 2 =
2

2 = Distance travelled in one complete orbit =
2 perimeter of orbit = 2
Linear speed of planet, =
2 ×
= 2


= 2 =

� �2
=

4 2 2
2 =

4 2 2 = × 2


4 2 2 × = 2


4 2 3 = 2


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2 = 4 2 3


2 = �4 2 � 3

o Kepler’s Third Law, ∝

compare, 2 ∝ 3
2 = 3

2 = �4 2 � 3

Constant, = 4 2


o Kepler’s Third Law is applied in:
(i) the system of planets and the Sun (M = mass of the Sun)
(ii) the system of satellites and the Earth (M = mass of the Earth)

4. Orbital period, or radius of orbit, can be calculated by 12 = 13 Period, 12 or 22 must
22 23 be the same unit

Kepler’s Third Law, 2 = �4 2 � 3

Comparison between two planets:

Planet 1: Planet 2:
( 1)2 = �4 2 � 1 3 ( 2)2 = �4 2 � 23

Planet 1 ÷ Planet 2:

( 1)2 = �4 2 � 1 3
( 2)2 = �4 2 � 23

1 2 = 1 3
2 2 23

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5. The application of the formula 12 = 2133:
22

(i) For the planets that orbiting the Sun: (as shown in Diagram 5)

o = distance between the center of the planet and the center of the

Sun

Diagram 5: distance between the center of the planets and the center
of the Sun, 1 dan 2

(ii) For the satellite orbiting the Earth: (as shown in Diagram 6)
o = distance between the center of the Earth and the center of
satellite
o = + ℎ
= radius of the Earth = 6.37 × 106 m
ℎ = height of the satellite from the center of the Earth

Diagram 6: A satellite at the height, ℎ from the surface of the Earth

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MIND MAP

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FAQ

No Question And Answer
1Q What are the different between Newton’s Laws and Kepler’s Laws?

A Newton’s Third Law are general and apply to any motion. While Kepler’s
2Q Law apply only to planetary motion in the solar system. It made detailed
measurements of the motions of the planets in the sky.
A
How are Newton’s Laws related to Kepler’s?
3Q
A Notice that (because of Kepler's 2nd Law) the velocity vector is
constantly changing both its magnitude and its direction as it moves
around the elliptical orbit (if the orbit were circular, the magnitude of
the velocity would remain constant but the direction would change
continuously). Since either a change in the magnitude or the direction of
the velocity vector constitutes an acceleration, there is a continuous
acceleration as the planet moves about its orbit (whether circular or
elliptical), and therefore by Newton's 2nd Law there is a force that acts
at every point on the orbit. Furthermore, the force is not constant in
magnitude, since the change in velocity (acceleration) is larger when the
planet is near the Sun on the elliptical orbit.

Kindly click to this link and see the illustrated in adjacent animation :
http://www.pas.rochester.edu/~blackman/ast104/newtonkepler.html

Does centripetal force affect gravity?

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Gravity along with all other fundamental forces (electromagnetic, strong
and weak) have effects on the centripetal force and nothing else can
affect it. Centripetal force is just a name given to the resultant of these
forces when a body has some form of rotational motion.
4 Q Why does a planet move slower when it is further from the sun?
A A planet moves slower when it is further from the Sun because its
angular momentum does not change. For a circular orbit, the angular
momentum is equal to the mass of the planet (m) times the distance of
the planet from the Sun (r) times the velocity of the planet (v). Since m x
v x r does not change, when a planet is close to the Sun, r becomes
smaller as v becomes larger. When a planet is far from the Sun, r
becomes larger as v becomes smaller.
5 Q Where is earth when it is travelling fastest rate?

A

It follows from Kepler’s second law that Earth moves the fastest when it
is closest to the Sun. This happens in early January, when Earth is about
147 million km (91 million miles) from the Sun. When Earth is closest to
the Sun, it is traveling at a speed of 30.3 km (18.8 miles) per second.
6 Q Does the Sun move around the Milky Way?

A

Yes, the Sun — in fact, our whole solar system — orbits around the
center of the Milky Way Galaxy. The earth is moving at an average
velocity of 828,000 km/hr. But even at that high rate, it still takes us
about 230 million years to make one complete orbit around the Milky
Way!”

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EXERCISES 3.2

SECTION A

1 Which of the following diagram shows the correct location of the focus, F of one
ellipse?
[Mengetahui]

2 Diagram 2 shows four positions for a planet which is orbiting the sun M in an ellipse
orbit.

Diagram 2 [Mengetahui]
The velocity of the planet is minimum at 11

AW
BX
CY
DZ

3 The diagram below shows two planets orbiting the sun. Planet P takes 2 years while
planet Q takes 3 years to orbit the sun.

Planet P

rp

rQ

Planet Q

What is the radius of the orbit of planet Q? [mengaplikasi kuantitatif]
[Given that radius of the orbit planet P = 220 x106km ]

A 251X 105 km
B 251 X 106 km
C 288 X 106 km
D 28.8 X 105 km

4 Satellite M is 4 times farther from a planet than satellite N. When the satellite M
takes 20 weeks to complete a full orbit around the planet, how long will the
satellite N to travel around the planet once?
[Mengaplikasi kuantitatif]
A 2 Weeks
B 2.5 Weeks
C 4 Weeks
D 4.5 Weeks

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5 The diagram shows a satellite orbiting the earth in a period 2 hours.

Satelit /Satellite
T = 2 jam / hour

Bulan /Moon
T = 27 hari/ Days

If the period of the moon orbiting the earth is 27 days and the radius of the orbit
moon is 3.72 X 105 km.

What is the height of the satellite from the earth? [menganalisis kuantitatif]
[Given that the radius of the earth is 6 370 km]

A 1515.8 km
B 65612.5 km
C 7885.8 km
D 59242.5 km

SECTION B
1. Diagram 1 shows a planet orbiting the Sun.

TRS=2 hours TPQ=2 hours

D
Diagram 1

a) State the Kepler’s First Law.
[Mengatahui]

…………………………………………………………………………………………………………………………………….

……………………………………………………………………………………………………………………………………

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b) Based on Diagram 1 , [menganalisis]

i) compare the time taken for the planet to move from point P to point Q

and from point R to point S.

…………………………………………………………………………………………………………………….

ii) compare the distance of RS and the distance of PQ. [menganalisis]

…………………………………………………………………………………………………………………….

iii) compare the linear speed of the planet between points PQ and RS.
[menganalisis]

…………………………………………………………………………………………………………………….

iv) state the relationship between the distance travelled and the linear speed

…………………………………………………………………………………………………………………….
[menganalisis]

c) State the physics law involved in question (b). [Mengetahui]

…………………………………………………………………………………………………………………….

2. Diagram 2 shows the planets, Earth and Mars, orbiting the Sun.

Diagram 2
Orbital period of Mars can be determined by comparing the orbit of Mars with the
orbit of the Earth.

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(a) What information is needed to determine the orbital period of Mars?
[mengetahui]

…………………………………………………………………………………………………………………………………….
(b) The radius of orbit of the Earth is 149.6 x 109 m, orbital period of the Earth is
365.2 days and the radius of orbit of Mars is 227.9 x 109 m. Calculate the orbital
period of Mars.

[Mengaplikasi kuantitatif]

3. Diagram 3 shows Satellite 1 needs to orbit the Earth in a period of 24 hours in line
with the Earth’s rotation.

Diagram 3
a) State Kepler’s Third Law.

[Mengetahui]
……………………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………………

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b) Calculate the height of the satellite from the surface of the Earth.
[Mengaplikasi]

c) Based on the answer in di (a) ,determine
i) the gravitational acceleration acting on the satellite
ii) the linear speed of the satellite when orbiting the Earth.
[Radius of orbit of the Moon = 3.83 X 108 m, mass of Earth = 5.97 X 1024 kg
orbital period of the Moon = 655.2 hours, radius of Earth = 6.37 X 106 m ]
[Mengaplikasi kuantitatif]

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TITLE link QR CODE
KEPLER'S LAWS https://youtu.be/vNpqNTFnrM0

KEPLER'S LAW OF PLANETARY https://youtu.be/N5a9npp0
MOTION

KEPLERS LAWS https://youtu.be/2aNC9kv0Ukc

INTRODUCTION TO KEPLER'S LAWS https://youtu.be/HJ1fHPzhkZM
OF PLANETARY MOTION.

REINFORCEMENT TEST https://drive.google.com/file/d
/1j7H0zJVwY2vP6hJr6CpPpKyu
D7979zbY/view?usp=sharing

ANSWER (REINFORCEMENT TEST) https://drive.google.com/file/d
/1sSZoRbWc8PSaf98tIZbUzED3
ua6vqnd_/view?usp=sharing

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ANSWER

EXERCISE 3.2

SECTION A

No. Answer
1B
2B
3C
4B
5A

SECTION B

Structured question

Question 1

a) All planets move in elliptical orbits with the Sun at one focus (Law of Orbits).
b) i) Same time taken.

ii) The distance of PQ is more than the distance of RS.
iii) The linear speed of the planet between points PQ is higher than RS.
iv) The higher the linear speed, the more the distance travelled.
c) Kepler’s Second Law

Question 2

(a) Radius of the orbit of the Earth, orbital period of the Earth and radius of the orbit
of Mars.

(b)

12 = 22
1 3 2 3

365. 22 = 22 109)3
(149.6 × 109)3 (227.9 ×

22 = 365. 22 × (227.9 × 109)3
(149.6 × 109)3

= 686.67 days

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Question 3

a) The square of the orbital period of any planet is directly proportional to the cube of
the radius of its orbit (Law of Periods).

b)
T12 = T22
r13 r23

( )242 = (655.2)2

r13 3.83×108 3
r = 4.224×107 m
height of satellite from earth = 4.224×107 − 6.37 ×106

= 3.58×107 m
= 3.58×104 km

c)(i)

g = GM
r2

= 6.67 ×10−11 × 5.97 ×1024
4.224 ×107 2
( )g

g = 0.221 m s-2

c)(ii)
v = 2πr
T
v = 2π (4.224 ×107 )
24 × 60 × 60
v = 3071.78 m s-1

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