Gravitational fields

Just some useful questions to practice

What is the total gravitational potential at a point where the orbit of a geostationary satellite crosses the line directly between the centres of the Earth and Moon?

Data:

• Average radius of Moon’s orbit = 384,400 km

• Radius of geostationary Earth orbit = 4.23 x 10⁷ m

• Mass of Earth = 6.0 x 10²⁴ kg

• Mass of Moon = 7.3 x 10²² kg

Use the equation 𝑉_𝑔 = −𝐺𝑀/𝑟 to calculate the change in potential in going from the Earth’s surface to the International Space Station.

Data:

• RE = 6400 km

• ME = 6.0 x 10²⁴ kg

• G = 6.67 x 10^-11 N m² kg^-2

• Average height of ISS above Earth = 383 km

Sketch a graph of gravitational potential (y-axis) vs distance (x-axis) from the surface of the Earth to the surface of the moon.

Data:

• Vg on surface of Earth = -63 MJkg^-1

• Vg on surface of moon = -3 MJkg^-1

• Potential is a scalar so two non-zero potentials CANNOT combine to give zero.

• Just because there is a point where forces are balanced DOESNT mean the potential is zero at that point.

A rocket moves from r₁ to r₂.

1. Write down its change in displacement

2. Write down its gravitational potential V_g1 at r₁

3. Write down its gravitational potential V_g2 at r₂

4. Express its change in gravitational potential V_g2-V_g1

1. r₂ - r₁

2. V_g1 = - GM / r₁

3. V_g2 = - GM / r₂

4. V_g2 - V_g1= - GM/r₂ - (-GM/r₁) = - GM(1/r₂ - 1/ r₁)

1. What is the GPE of the ISS at the Earth’s surface?

2. What is the GPE of the ISS in its orbit?

3. What is the minimum energy required to place it in orbit?

4. Why is this value a minimum?

Data:

• Mass of ISS = 420 tonnes (1 tonne = 1000 kg)

• Mass of Earth = 6.0 x 1024 kg

• Radius of Earth = 6400 km

• Average height of ISS above Earth = 383 km

1. -2.63 x 10¹³ J

2. -2.48 x 10¹³ J

3. 1.5 x 10¹² J

4. This is a minimum because the satellite will also need kinetic energy for this orbit

A satellite of mass 200 kg has a velocity of 5.6 x 10³ ms^-1 and is orbiting at a distance of 1.28 x 10⁷ m from the centre of the Earth. The Earth has mass 6.0 x 10²⁴ kg. Calculate the KE, GPE and total energy of the satellite.

KE = 3.14 x 10⁹ J

GPE = - 6.25 x 10⁹ J

Total energy = - 3.12 × 10⁹ J

Calculate the escape velocity for Earth

Data:

• Mass of Earth = 6.0 x 10²⁴ kg

• G = 6.67 x 10^-11 N m² kg^-2

• Radius of Earth = 6400 km

v = 1.12 x 10⁴ m s^-1