International A LevelPhysicsEdexcelTopic Questions5. Thermodynamics, Radiation, Oscillations & CosmologyGravitational Fields

Gravitational Fields (Edexcel International A Level Physics)

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In the 17th century, Kepler proposed his ‘law of harmonies’ for planetary motion. This law suggested that the ratio of the square of the orbital period T to the cube of the mean radius R has the same value for all the planets that orbit the Sun.

Mathematically his ‘law of harmonies’ can be written

         T squared equals K R cubed

where K is a constant.

Kepler’s law of harmonies was derived later by Newton. Newton applied his law of gravitation to a planet moving in an approximately circular orbit around the Sun. 

Determine a value for K by applying Newton’s law of gravitation to a planet of mass m moving in a circular orbit about the Sun. 

   mass of Sun = 1.99 × 1030 kg





K = .............................. s2 m–3

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2
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In 2015, the company SpaceX stated its plan to launch about 4,000 “Starlink” satellites into a low Earth orbit. These satellites would provide a low-cost internet service to people around the world.

By the start of 2020 almost 200 Starlink satellites had been launched into an orbit 550 km above the Earth’s surface.

   mass of Earth = 6.0 × 1024 kg
   radius of Earth = 6.4 × 106 m

Calculate the orbital time of a Starlink satellite.




Orbital time of satellite = ...........................................

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1a
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In the 17th century, Kepler proposed his ‘law of harmonies’ for planetary motion. This law suggested that the ratio of the square of the orbital period T to the cube of the mean radius R has the same value for all the planets that orbit the Sun.

Mathematically his ‘law of harmonies’ can be written

         T squared equals K R cubed

where K is a constant.

The table shows data for Earth and Mars.

 
Planet T / s R / m
Earth 3.16 × 107 1.50 × 1011
Mars 5.93 × 107 2.28 × 1011

  
 Show that, for Earth and for Mars, K has a value of about 3.0 × 10–19 s2 m–3.

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1b
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Jupiter is the most massive planet in the solar system and has many moons. Kepler’s law of harmonies applies to the orbiting moons. However, the value of K for the moons is not the same as the value that applies to planets orbiting the Sun.

Ganymede is Jupiter’s largest moon. Ganymede has an orbital radius of 1.07 × 109 m and an orbital period of 172 hours.

Another moon, Io, has an orbital radius of 4.22 × 108 m.

Calculate the orbital period TI of Io about Jupiter.

TI= ............................

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2a
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Weather satellites may be in an equatorial orbit or a polar orbit, as shown.

q21-wph15-05-oct-2021-edexcel-int-as-and-a-level-phy

A weather satellite in a polar orbit circles the Earth at a height of 8.50 × 105 m above the surface of the Earth.

i)
Show that the gravitational potential at this height is about −5.5 × 107 J kg−1.
mass of Earth = 5.98 × 1024 kg
radius of Earth = 6360 km

(2)

ii)
Hence calculate the increase in gravitational potential energy when the satellite is placed in orbit.

satellite mass = 4990 kg
gravitational potential at the Earth’s surface = −6.27 × 107 J kg−1

(2)



Increase in gravitational potential energy = .........................................

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2b
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It is claimed that a satellite in orbit at a height of 8.50 × 105 m would make 15 orbits of the Earth every 24 hours.
Assess the validity of this claim.

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2c
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It is suggested that the satellite could be placed in an equatorial orbit at a radius of 42 200 km. With this radius the satellite would have an orbital period equal to 24 hours.
Give one advantage and one disadvantage of placing a weather satellite in a polar orbit rather than in the suggested equatorial orbit.

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3a
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In 2015, the company SpaceX stated its plan to launch about 4,000 “Starlink” satellites into a low Earth orbit. These satellites would provide a low-cost internet service to people around the world.

By the start of 2020 almost 200 Starlink satellites had been launched into an orbit 550 km above the Earth’s surface.

   mass of Earth = 6.0 × 1024 kg
   radius of Earth = 6.4 × 106 m

Calculate the orbital time of a Starlink satellite.



Orbital time of satellite = ...........................................

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3b
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In 2015, the company SpaceX stated its plan to launch about 4,000 “Starlink” satellites into a low Earth orbit. These satellites would provide a low-cost internet service to people around the world.

By the start of 2020 almost 200 Starlink satellites had been launched into an orbit 550 km above the Earth’s surface.

   mass of Earth = 6.0 × 1024 kg
   radius of Earth = 6.4 × 106 m

Explain why satellites in low Earth orbits have a smaller orbital time than satellites in higher Earth orbits.

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3c
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The satellites were launched into orbit using rockets able to make multiple space flights.
Calculate the minimum kinetic energy required to raise the satellite to its lower orbit height.
   mass of a Starlink satellite = 227 kg.






Minimum kinetic energy = ......................................

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