Presentation on theme: "9.2 Gravitational field, potential and energy"— Presentation transcript:
19.2 Gravitational field, potential and energy 9: Motion in Fields9.2 Gravitational field, potential and energy
2GravityRecap:Newton’s universal law of gravitation:Gravitational field strength:F = GMm r2…the force per unit mass experienced by a small test mass (m) placed in the field.g = GMr2
3GPE in a uniform fieldWhen we do vertical work on a book, lifting it onto a shelf, we increase its gravitational potential energy (Ep). If the field is uniform (e.g. Only for very short distances above the surface of the Earth) we can say...GPE gained (Ep) = Work done = F x d= Weight x Change in heightso ΔEp = mg∆hE.g. In many projectile motion questions we assume the gravitational field strength (g) is constant.
4GPE in non-uniform fields However, as Newton’s universal theory of gravity says, the force between two masses is not constant if their separation changes significantly. Also, the true zero of GPE is arbitrarily taken not as Earth’s surface but at ‘infinity’.‘Infinity’Ep = 0Lots of positive work must be done on the small mass!Ep = negativeIf work must be done to “lift” a small mass from near Earth to zero at infinity then at all points GPE must be negative. (This is not the same as change in GPE which can be + or -)
5Strictly speaking, the GPE is thus a property of the two masses. The GPE of any mass will always be due to another mass (after all, what is attracting it from infinity?)Strictly speaking, the GPE is thus a property of the two masses.E.g. Calculate the potential energy of a 5kg mass at a point 200km above the surface of Earth.( G = 6.67 N m2 kg-2 , mE= 6.0 1024 kg, rE= 6.4 106 m )The gravitational potential energy of a mass at any point is defined as the work done in moving the mass from infinity to that point.Ep = - GMm r
6The gravitational potential energy of a mass at any point is defined as the work done in moving the mass from infinity to that point.
7Q. What do the indicated properties of these two graphs represent? b
9Gravitational Potential Whereas gravitational force on an object on Earth depends upon the mass of the object itself, gravitational field strength is a measure of the force per unit mass of an object at a point in Earth’s field.Similarly, whereas the GPE of say a satellite, depends upon both the mass of Earth and the satellite itself, gravitational potential is a measure of the energy per unit mass at a point in Earth’s field.
10Thus for a field due to a (point or spherical) mass M: So ...E.g. Calculate the potential of a 5kg mass at a point 200km above the surface of Earth. What would be the potential of a 10kg mass at the same point?( G = 6.67 N m2 kg-2 , mE= 6.0 1024 kg, rE= 6.4 106 m )The gravitational potential at a point in a field is defined as the work done per unit mass in bringing a point mass from infinity to the point in the field.V = Ep = - GMmm r mV = Gravitational potential (Jkg-1)V = - GM r
11Gravitational Potential in a uniform field. For a uniform field…∆Ep = mg∆hSo…∆V = ∆Ep = mg∆hm m∆V = g∆h
12How far apart are the equipotentials in this diagram?
14Equipotential Surfaces Equipotential surfaces or lines join points of equal potential together. Thus if a mass is moved around on an equipotential surface no work is done.Thus the force due to the field, and therefore the direction of the field lines, must be perpendicular to the equipotential surfaces at all times.
15Potential GradientThe separation of the equipotential surfaces tells you about the field:Uniform fields have equal separationFields with decreasing field strength have increasing separation.
16If the equipotentials are close together, a lot of work must be done over a relatively short distance to move a mass from one point to another against the field – i.e. the field is very strong. This gives rise to the concept of ‘potential gradient’.The ‘potential gradient’ is given by the formula...Potential gradient = ΔVΔrIt is related to gravitational field strength...g = - ΔV
17Escape speed So... but… so… Loss of KE = Gain in GPE If a ball is thrown upwards, Earth’s gravitational field does work against it, slowing it down. To fully escape from Earth’s field, the ball must be given enough kinetic energy to enable it to reach infinity.Loss of KE = Gain in GPE½ mv2 = GMm (Note this also = Vm)rSo but… so…The escape speed is the minimum launch speed needed for a body to escape from the gravitational field of a larger body (i.e. to move to infinity).
18Note we could also say...½ mv2 = GMm = VmrSo... v = √(2V)
19Note we could also say...½ mv2 = GMm = Vmrso... v = √(2V)Assumptions…Planet is a perfect sphereNo other forces other than gravitational attraction of the planet.Note:Applies only to projectiles- Direction of projection is not important if we assume that the planet is not rotating