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Gravitation © David Hoult 2009
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© David Hoult 2009
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© David Hoult 2009
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© David Hoult 2009
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F a m1m2 © David Hoult 2009
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© David Hoult 2009
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1 F a r2 © David Hoult 2009
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m1m2 F = G r2 where G is the universal gravitation constant
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m1m2 F = G r2 N m2 kg-2 where G is the universal gravitation constant
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Testing the Inverse Square Law of Gravitation
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9.8 = × 10-3 ms-2 602 © David Hoult 2009
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9.8 = × 10-3 ms-2 602 v2 a = r © David Hoult 2009
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9.8 = 2.72 × 10-3 ms-2 602 v2 a = r r = 3.84 × 108 m T = 27.3 days
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Centripetal acceleration of the moon (caused by the force of gravity)
2.72 × 10-3 ms-2 © David Hoult 2009
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The inverse square law is a good theory
Conclusion The inverse square law is a good theory © David Hoult 2009
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Relation between g and G
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Relation between g and G
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Relation between g and G
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Relation between g and G
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we have assumed the equivalence of inertial and gravitational mass
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Gravitational Field Strength
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The g.f.s. at a point in a gravitational field is the force per unit mass acting on point mass
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The g.f.s. at a point in a gravitational field is the force per unit mass acting on point mass
Units Nkg-1 © David Hoult 2009
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“Force per unit mass” is equivalent to acceleration
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G.f.s. is another name for acceleration due to gravity
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1 g a r2 © David Hoult 2009
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1 g a r2 © David Hoult 2009
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© David Hoult 2009
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1 outside the sphere g a r2
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1 outside the sphere g a r2 © David Hoult 2009
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1 outside the sphere g a r2 inside the sphere g a r © David Hoult 2009
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1 outside the sphere g a r2 inside the sphere g a r © David Hoult 2009
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© David Hoult 2009
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© David Hoult 2009
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World High Jump Record... © David Hoult 2009
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World High Jump Record... on Mars ? © David Hoult 2009
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maximum height, s depends on:
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maximum height, s depends on:
initial velocity, u © David Hoult 2009
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maximum height, s depends on:
initial velocity, u acceleration due to gravity, g © David Hoult 2009
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so, for a given initial velocity
u2 = -2gs so, for a given initial velocity © David Hoult 2009
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so, for a given initial velocity
u2 = -2gs so, for a given initial velocity gs = a constant © David Hoult 2009
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For a given initial velocity, the maximum height reached by the body is inversely proportional to the acceleration due to gravity © David Hoult 2009
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1 s a g © David Hoult 2009
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1 s a g sg = a constant © David Hoult 2009
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1 s a g gs = a constant g1s1 = g2s2 or s1 g2 = s2 g1
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Gravitational Potential
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The potential at a point in a gravitational field is the work done per unit mass moving point mass from infinity to that point © David Hoult 2009
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units of potential J kg-1
The potential at a point in a gravitational field is the work done per unit mass moving point mass from infinity to that point units of potential J kg-1 © David Hoult 2009
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w = Fs but in this situation the force is not of constant magnitude
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w = Fs but in this situation the force is not of constant magnitude
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It is clear that the work done will depend on:
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It is clear that the work done will depend on:
the mass of the planet, M © David Hoult 2009
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It is clear that the work done will depend on:
the mass of the planet, M the distance, r of point p from the planet © David Hoult 2009
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It is clear that the work done will depend on:
the mass of the planet, M guess: w a M the distance, r of point p from the planet © David Hoult 2009
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It is clear that the work done will depend on:
the mass of the planet, M guess: w a M the distance, r of point p from the planet guess: w a 1/r © David Hoult 2009
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...it can be shown that... © David Hoult 2009
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GM w = r © David Hoult 2009
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“at infinity” means that the body is out of the gravitational field
A body at infinity, has zero gravitational potential “at infinity” means that the body is out of the gravitational field © David Hoult 2009
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All bodies fall to their lowest state of potential (energy)
A body at infinity, has zero gravitational potential “at infinity” means that the body is out of the gravitational field All bodies fall to their lowest state of potential (energy) © David Hoult 2009
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A body at infinity, has zero gravitational potential
“at infinity” means that the body is out of the gravitational field All bodies fall to their lowest state of potential (energy) All gravitational potentials are therefore negative quantities © David Hoult 2009
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GM V = r © David Hoult 2009
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GM V = r Therefore the gravitational potential energy possessed by a body of mass m placed at point p is given by © David Hoult 2009
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GM V = r The gravitational potential energy possessed by a body of mass m placed at point p is given by G P E = V m © David Hoult 2009
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Escape Velocity © David Hoult 2009
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G P E = zero © David Hoult 2009
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G P E = zero To find the minimum velocity, ve which will cause the rocket to escape the Earth’s gravity, assume K E of distant rocket is also equal to zero. © David Hoult 2009
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G P E = zero To find the minimum velocity, ve which will cause the rocket to escape the Earth’s gravity, assume K E of distant rocket is also equal to zero. As the body is moving away from the planet it is losing K E and gaining G P E © David Hoult 2009
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G P E = zero To find the minimum velocity, ve which will cause the rocket to escape the Earth’s gravity, assume K E of distant rocket is also equal to zero. As the body is moving away from the planet it is losing K E and gaining G P E D K E = D G P E © David Hoult 2009
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If the mass of the rocket is m, then the G P E it possesses at the surface of the planet is
GMm G P E = R © David Hoult 2009
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If the mass of the rocket is m, then the G P E it possesses at the surface of the planet is
GMm G P E = R GMm D G P E = r © David Hoult 2009
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GMm G P E = R GMm D G P E = R ½mve2
If the mass of the rocket is m, then the G P E it possesses at the surface of the planet is GMm G P E = R GMm D G P E = R D K E = ½mve2 © David Hoult 2009
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GMm ½mve2 = R © David Hoult 2009
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Also, as g = GM/R2 © David Hoult 2009
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Also, as g = GM/R2 © David Hoult 2009
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