topic 12 gravitational fields

• Gravitational field strength


  Force per unit mass
• Close to the Earth’s surface, the gravitational field strength is uniform and approximately equal to the acceleration of free fall
• Period of an object describing a circle

  The time taken for an object to complete one full revolution around a circle
• Geostationary orbit of a satellite


  An orbit where a satellite remains above the same point on the Earth's surface
• Mass of an object creates a gravitational field around it, exerting an attractive force on any other mass in the field region
• When an object is dropped, the Earth and the object exert equal and oppositely directed forces on each other, but the object is pulled towards the Earth due to its smaller mass
• Weight of an object


  Force of gravity acting on it
• If an object of mass (m) is in a gravitational field of strength (g), the gravitational force (F) on the object is F = mg
• If the object is allowed to fall freely under the action of this force, it accelerates with an acceleration a = g
• If the object is allowed to fall freely under the action of this force, it accelerates with an acceleration: a = F = mg = g
• The average value of the Earth’s gravitational field strength is 9.81 N kg-1
• Gravitational field strength


  Field strength at any point = The acceleration of free fall in a gravitational field (N kg-1) experienced by an object at that point (m s-2)
• Show that N kg-1 is the same as m s-2
• Field lines enable us to picture the shape of the field as well as the direction of the forces around the body
• The direction of the field lines indicates the direction of the gravitational force acting on a mass situated in the field
• The field lines are directed towards the centre of the planet, indicating that the gravitational field is attractive
• In a radial field, the separation of the field lines increases with distance from the centre, indicating that the field strength is decreasing as the distance increases
• Close to the surface and over an area small in comparison with the overall area of the planet, the field can be assumed to be uniform (i.e. constant strength and direction), indicated by parallel field lines
• Consider two point masses (m1 and m2) whose centres are distance (r) apart. Then, using Newton’s law of gravitation, the gravitational attraction force (F) which each mass exerts on the other is given by: F α m1 m2 / r^2
• Newton’s law of gravitation


  • Every particle in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of their separation
• The minus sign in the equation is there because it is conventional in field theory to regard forces exerted by attractive fields as negative, and gravity is attractive everywhere in the universe. Another reason is that r is measured outwards from the attracting body and F acts in the opposite direction
• Gravitational forces are extremely weak, unless at least one of the objects is of planetary mass or larger
• Gravitational forces act at a distance, without the need for an intervening medium
• Newton’s law is expressed in terms of point masses. For real bodies, the law can be applied by assuming all the mass of a body to be concentrated at its centre of mass. The separation (r) is then the distance between the centres of mass
• Gravitational constant (G)


  6.67 x 10^-11 N m^2 kg^-2
• Newton’s law of gravitation is an example of an inverse square law
• Calculating gravitational force


  Using the formula F = - G m1 m2 / r^2
• Definition of field strength

  The force (F) acting on (m) is F = mg
• Applying Newton’s law of gravitation


  The force (F) is F = -GMm/r^2
• Combining equations


  mg = -GMm/r^2
• Below the surface: gravitational field strength (g) is directly proportional to distance from the centre of the Earth (r)
• At the centre: gravitational field strength (g) = 0
• For r > R (Earth radius): gravitational field strength (g) is inversely proportional to r^2
• All the above applies to any planet or star
• Show that the gradient is equivalent to GM, where G is the universal gravitational constant, and M is the mass of the Moon
• Determine the mass (M) of the Moon


  Calculate the mass of the Moon using the gradient
• Use the internet to find the surface gravitational field strength and the diameter of the planets in the solar system
• Calculate the mass of each planet

  Calculate the mass of each planet using the data obtained
• Check the calculated values using the internet
• Any body orbiting a planet is a satellite of that planet