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Transcript of Gravitational Fields
The mass of an object creates a gravitational field around it. Gravitational Fields Gravitational Field Strength There is gravitational attraction between all objects. Gravitational field strength, g, is the force per unit mass acting on a small test mass in a gravitational field. If a small test mass is placed near to a very large mass they will attract each other with equal and opposite forces... (From Newton's third law of motion) But only the small mass will noticeably move towards the other, due to the difference in their sizes. The path which a test mass will follow towards a massive body is called a field line or line of force Field Lines Radial Fields Field lines act towards centre of mass. Earth's gravitational field is radial overall... Uniform Fields Field lines are parallel. Earth's gravitational field is uniform on a small scale... The increase in distance between field lines as distance from the planet increases is negligible. g=F/m acceleration = F/m so a=g In order to move away from Earth, a rocket must do work to overcome gravity. Gravitational Potential Gravitational potential energy is the energy of an object due to its position in a gravitational field. As the rocket rises, its gravitational potential energy increases and it does work. Gravitational Potential The gravitational potential, V, at a point is the work done per unit mass to move a small object from infinity to that point. Gravitational potential energy (gpe) is zero at infinity by convention. Hence, gpe is always a negative value. V is also zero at infinity and negative towards the planet's surface. Because V and gpe are different at the surfaces of different planets. V=W/m gpe=mV Potential gradients Equipotentials are lines of equal gravitational potential around a body (in red in diagram): Notice that at increasing distances from the planet's surface, the equipotentials are spaced further apart. This is because the gravitational field is becoming weaker. The potential gradient at a point in a gravitational field is the change in gravitational potential per metre at that point. Potential gradient = change in V/change in r: r = distance moved The closer the equipotentials are together, the greater the potential gradient is and the stronger the gravitational field is. g = -(potential gradient) Kepler's third law: Newton's Law of Gravitation For a system (e.g. all the planets round the sun): = a constant Newton realised that there is a force of gravitational attraction between the sun and the planets that caused them to orbit it. He assumed that this force was inversely proportional to the square of their distance apart. Inverse square law: If the distance doubles, the force is divided by 4. If the distance is tripled, the force is divided by 9. If the distance is quadrupled, the force is divided by 16. Newton's law of gravitation assumes that the gravitational force between any two point objects is: Always an attractive force
Proportional to the mass of each object
Inversely proportional to the square of their distance apart. Cavendish's measurement of G: G=gravitational constant (Big G) How the measurement was made: The wire from which the small lead balls were suspended was calibrated by measuring the couple needed to twist it by one degree. Then, the two large lead balls were brought near to the small ones. The gravitational attraction between the large and small balls caused the small balls to move, twisting the wire. By measuring the angle through which the wire had twisted, the force of gravitational attraction between the large and small balls could be calculated. From this, big G could be derived. Equations Satellite Motion Gravitational Field Strength Planetary Fields and Therefore Where m1 is the mass of the planet. Variation of G with distance (From a sherical planet or star) R 2R g 1/4 g 1/9 g 3R Inverse Square Law is the reason for the shape after R (Radius - this is the surface of the planet). Inside the planet, g decreases with decreasing r. This is because only the mass of the planet that is within r at that point, contributes to g. Gravitational Potential Near a Spherical Planet Potential Gradients Near a Spherical Planet V R This is a -(1/r) graph r Assuming the centripetal force keeping a satellite moving in a circle is the gravitational force, Therefore, as F=ma (so a=F/m), Using the formula for centripetal acceleration (see circular motion chapter), Geostationary Satellites Orbit Earth directly above the equator. Have a time period of 24h, and therefore remain in a fixed position relative to the Earth. Used for communication, SATNAVs etc. The end.