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D.1 Gravitational Fields

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2024/07/05 08:44
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D.1.1 Newton's law of gravitation

Newton’s law of gravitation
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Gravitational force between two objects can be calculated using Newton’s universal law of gravitation
F=Gm1m2r2G=gravitational constant=6.67×10−11 m1=mass of object 1,m2=mass of object 2,r=distance between objectsF = \frac{G m_1 m_2}{r^2}\\ G = \textit{gravitational constant} = 6.67 \times 10^{-11}\\  m_1 = \textit{mass of object 1}, \quad m_2 = \textit{mass of object 2}, \quad r = \textit{distance between objects}
D.1.1-1 Diagram with equation explaining the gravitational force
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Newton’s universal law of gravitation states that every object attracts other objects with a force
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It is directly proportional to the product of the masses
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It is inversely proportional to the square of the distance between the objects (It obeys the inverse square law
D.1.1-2 simplified graph explaining the relationship between the gravitational force and distance between objects

D.1.2 Gravitational field strength

Gravitational field strength
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Gravitational field strength at a point is the force per unit mass experienced by a test mass at that point
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The gravitational field strength due to an object is :
g=Fm g=Gravitational field strength,F=gravitational force,m=massg = \frac{F}{m}\\  g = \textit{Gravitational field strength}, \quad F = \textit{gravitational force}, \quad m = \textit{mass}
D.1.2-1 Graph of gravitational potential over distance between the objects
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When throwing a ball high above from the surface of the Earth, the ball travels in an orbit.
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Newton proposed an imaginary cannon experiment high above the ground, that releases a ball in an orbit
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Centripetal force applied to the ball is equal to the gravitational force
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There is an equilibrium of forces, which makes the ball maintain its height
D.1.2-2 Diagram of the Newton’s cannon
Fc=FG=mv2r=GMEmr2F_c = F_G = \frac{mv^2}{r} = \frac{GM_E m}{r^2}
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From this equation, we can derive the orbital speed:
v=GMErv = \sqrt{\frac{GM_E}{r}}
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Revisiting topic 4, this can be converted into the equation of angular speed,
ω2=GMEr3\omega^2 = \frac{GM_E}{r^3}
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Kepler’s 3rd law uses this founding from equations above, which defines the relationship between the period of the planet and its radius:
T2=4π2r3GMET^2 = \frac{4\pi^2 r^3}{GM_E}

D.1.3 Gravitational Fields

About Field
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General definition of field is :
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A region or area which an object experiences a specific force related to the origin of region in influence of distance
D.1.3-1 Diagram with equation of field
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Field is shown in various areas and two main areas that construct field in every situation are :
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Gravitational Field
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Electrostatic Field
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Gravitational Field :
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The gravitational force per unit mass exerted on a point mass
D.1.3-2 Line diagram of gravitational field
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Field can be measured by the magnitude of strength
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We call it field strength
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Field strength can be calculated by dividing the force acting on a test object by the size of the test object
Field strength=force acting on a test objectsize of test object\textit{Field strength} = \frac{\textit{force acting on a test object}}{\textit{size of test object}}
Uniform Fields
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A gravitational field is a region of space in which objects with mass will experience a force
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The gravitational field strength can be calculated using the equation :
g=Fm(g=gravitationalfieldstrength,F=gravitationalforce,m=mass)g=\frac Fm \\ (g=gravitational field strength, F=gravitational force, m=mass)
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Gravitational force is always attractive towards both objects
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It is important that gravitational field strength is always measured with the test point mass
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Which means that we are assuming the mass has no volume
D.1.3-3 Simple diagram of gravitational field between two masses
Radial Fields
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A point charge and point mass produce a radial field around them
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In calculation of magnitudes in radial field the charge or mass of two interacting masses or charges and distance between them are relative
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Inverse Square Law : the field strength decreases by a factor of inverse square of distance
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For Gravitational Field force between two masses :
FG=GMmr2(Fg=gravitational force,G=Newton’s gravitational constant)(M/m=masses of two point masses,r=distance between two masses)F_G = \frac{G M m}{r^2}\\ (F_g = \textit{gravitational force}, \quad G = \textit{Newton's gravitational constant})\\ (M/m = \textit{masses of two point masses}, \quad r = \textit{distance between two masses})

D.1.4 Gravitational Field Lines

Gravitational Field Lines
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The direction of a gravitational field is represented by gravitational field lines
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The gravitational field lines around a point mass are radially inwards
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The gravitational field lines of a uniform field, where the field strength is the same at all points, is represented by equally spaced parallel lines
D.1.4-1 Annotated diagram of point mass and surface mass

D.1.5 Gravitational Potential

Gravitational Potential
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Gravitational potential energy : the energy an object possess due to its position in a gravitational field
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Gravitational potential at a point can also defined as :
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The work done per unit mass in bringing a test mass from infinity to a point
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This definition leads to the fact that the gravitational potential on an object will be zero when the distance between the object and mass is infinity
D.1.5-1 Diagram of gravitational potential earth
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The gravitational potential at a point depends on :
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The mass of object that provides the field
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The distance between the centre of mass and the point
D.1.5-2 Diagram of satellite of earth
Calculating Gravitational Potential
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The gravitational potential can be calculated :
V=−GMr(V=gravitational potential,G=Newton’s gravitational constant) (M=mass of the field provider,r=distance between mass and the point mass)V = -\frac{G M}{r}\\ (V = \textit{gravitational potential}, \quad G = \textit{Newton's gravitational constant})\\  (M = \textit{mass of the field provider}, \quad r = \textit{distance between mass and the point mass})
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Gravitational potential always is negative near an isolated mass
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The potential when r is at infinity is zero
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Work must be done to move a mass away from a planet

D.1.6 Potential Gradient & Difference

Gravitational Potential Gradient
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A gravitational field can be defined in terms of the variation of gravitational potential at different points in the field :
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The gravitational field at a particular point is equal to the negative gradient of a potential-distance graph at that point
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The potential gradient in a gravitational field is defined as :
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The rate of change of gravitational potential with respect to displacement in the direction of the field
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Gravitational field strength, g and the gravitational potential
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Gravitatioal potential can be graphically represented against the distance from the centre of a planet, r :
g=−ΔVΔr (g=gravitational field strength,ΔV=change in gravitational potential,Δr=distance)g = -\frac{\Delta V}{\Delta r}\\  (g = \textit{gravitational field strength}, \quad \Delta V = \textit{change in gravitational potential}, \quad \Delta r = \textit{distance})