Gravitational Force and Planetary Motion

Describe a gravitational field in the region surrounding a massive object in terms of its effect on other masses in it

A gravitational field provides a force on objects within it that drags objects to the centre of the field.

The strength of the field is related to the mass of the object that produces it, with larger masses resulting in stronger fields. A massive object will have a strong gravitational field that will attract other masses near it. If these masses have little or no tangential velocity, they will be dragged into the massive object. If they have some degree of tangential velocity, they will be pulled into orbit, or they will have their trajectory through space altered by the massive object with the force acting on the object pulling it towards the massive object.

Remember- A massive object has a gravitational field that drags other masses towards it.

Define Newton’s Law of Universal Gravitation

Newton’s Law of Universal Gravitation provides a formula by which the force exerted by gravity in a field can be calculated based on the masses involved and the distance between them. Gravitational force is equal to the multiple of the masses of the two objects, divided by the distance between them squared, then multiplied by the gravitational constant. F = \frac{Gm_1m_2}{d^2}. This formula serves to calculate the force experienced each of the bodies- however, the body with the larger mass will be less affected, because according to, F = ma if F is constant and m is large, then acceleration must be small.

Remember – Universal gravitation calculates the force experienced by each of the objects, and is experienced by both of them equally.

Present information and use available evidence to discuss the factors affecting the strength of gravitational force

There are numerous factors affecting the strength of gravity on Earth. Firstly, as the Earth spins it bulges at the equator, flattening at the poles. This causes the poles to be closer to the centre of the Earth than the equator. According to the formula for gravitation force, the force experienced depends on the distance from the centre of the field. This means that Earth’s gravitational field is stronger at the poles than at the Equator. Secondly, the field of the Earth varies with the density of nearby geography. Places where the lithosphere is thick, or where there are dense mineral deposits or nearby mountain experience greater gravitational force compared to places over less dense rock or water. Thirdly, as gravitational force depends on altitude, places with greater elevation such as mountain ranges experience less gravitational force, than areas at or below sea level. Finally, and more generally, gravitational force also depends on the mass of the central body, so that planets or bodies with less mass have weaker gravitational fields and therefore weaker gravitational force.

Remember- The Earth’s gravitational field is changed by distance from the equator, altitude, and lithosphere composition.

Discuss the importance of Newton’s Law of Universal Gravitation in under- standing and calculating the motion of satellites

In order to launch a satellite, the orbital velocity required must be known. The centripetal force acting on a body in orbit must be equal to the force that gravity exerts in order to keep the body in orbit. This means

F_c = F_g

and therefore

\frac{Gm_pm}{r^2}=\frac{mv^2}{r}

where m_p is the mass of the planet, and m is the mass of the satellite. Simplifying this expression yields

v=\sqrt{\frac{Gm_p}{r}}

Since Newton’s Law of Universal Gravitation is required to quantify the value of Fg in the derivation of orbital velocity (and indeed in any calculation involving gravitational field strength), it is therefore vital to understanding and calculating the motion of satellites. Further, Newton’s Law can be used to derive Kepler’s Law of Periods, an integral tool in understanding the motion of satellites in a given system. So although it is by no means a complete solution to understanding orbital motion, it is nonetheless an integral tool.

Remember- Newton’s Law of Universal Gravitation is vital to mathematically modelling orbits, and was used to derive Kepler’s Law of Periods.

1.3.5    Identify that a slingshot effect can be provided by planets for space probes

Note that some resources have the probe approaching the planet from the front, i.e. against the planet’s orbital direction. This also provides the same slingshot effect, but it is harder to visualise and understand.

If trajectories are calculated carefully, space probes can use the motion of planets through space in order to increase the probe’s velocity. In order to take advantage of the slingshot effect, the space probe approaches a planet in the same direction as the planet’s orbital path i.e. it approaches the planet from behind. When the probe enters the field, the probe is accelerated. However, the field itself is moving at the same time, because the planet is moving. This additional momentum is also given to the probe, as the probe is effectively dragged by the planet. When the probe leaves the gravitational field, the momentum it gained simply by falling into the field is lost (since it is climbing up and out of the gravitational field). However, the momentum gained by the dragging effect is retained, boosting the velocity of the probe. This is the slingshot effect- using the motion of planets to accelerate space probes. Another application of the slingshot effect is the altering of trajectory. For a probe to travel to the outer planets, it must travel away from the sun. However, the energy required to leave the sun’s gravitational field is immense. The probe’s trajectory outwards is gradually curved into an orbital path by the sun’s gravity. Using a variation of the slingshot effect, the probe can use a planet’s gravitation field not to gain velocity, but to alter its trajectory away from the sun. Ordinarily this trajectory change would consume large amounts of fuel, but the harnessing of the motion of planets removes this need, as well as reducing the time taken for a probe to visit the outer planets.

figure 9

 

Remember- The slingshot effect uses the movement of planets to change a space probe’s speed or direction to help it reach outer planets.