7 Gravitational Field Flashcards
State Newton’s Law of Gravitation and the equation to calculate the magnitude of gravitational force F between two particles of masses M and m which are separated by a distance r.
State the value of the gravitational constant and whether gravitational force is a vector or scalar quantity.
Newton’s Law of Gravitation states that two point masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. (SI unit: N)
The magnitude of gravitational force F between two particles of masses M and m which are separated by a distance r is given by, F = GMm/r^2 where G is the gravitational constant with a value of 6.67*10^-11 Nm^2kg^-2.
It is a vector quantity.
Is Newton’s Law of Gravitation only applicable to point masses? What characteristics do the gravitational forces between two point masses have?
Yes, every spherical object with constant density can be considered as a point mass at the centre of the sphere. The gravitational forces are attractive in nature and they are equal and opposite, constituting an action-reaction pair. These two forces always act along the line joining the two point masses.
Define gravitational field.
A gravitational field is a region of space in which a mass placed in that region experiences a gravitational force. (It consists of an array of imaginary field lines with the direction being equal to the direction of the gravitational field. For a point mass or uniform spherical mass, the field lines are directed towards its centre and are perpendicular to the surface. Near the surface of the earth, the gravitational field is approximately uniform as the field lines seem to be parallel to each other and evenly spaced. This is because the density of the field lines (number of lines per unit area) indicates its strength. This also means that field is stronger nearer the centre.)
Define gravitational field strength. State the formula for the gravitational field strength due to a point mass. State the relationship between gravitational field strength and distance r from the centre of the uniform solid sphere inside and outside of the sphere. Explain what this means for a hollow sphere.
The gravitational field strength at a point in space is defined as the gravitational force experienced per unit mass at that point. Formula: g = F/m
Inside a uniform solid sphere, the gravitational field strength die to the inner sphere varies linearly with distance r (from shell theorem: sub 4pir^3/3 into the formula above). However, outside a uniform solid sphere or hollow sphere, the gravitational field strength is identical to that of a point mass at the centre of the sphere and g varies inversely with r^2.
Define gravitational potential. State the formula for gravitational potential due to a point mass.
Gravitational potential at a point in a gravitational field is defined as the work done per unit mass by an external force in bringing a small test mass from infinity to that point. (Small test mass: So that no other gravitational field strength affects gravitational potential) (pg 10) Formula: phy = -GM/r (SI unit = J kg^-1)
Define gravitational potential energy. Explain what is meant by ‘a point in infinity’. State the formula for gravitational potential energy U for two particles of masses M and m separated by a distance r. Is U a scalar or vector quantity?
The gravitational potential energy of a mass at a point in a gravitational field is defined as the work done by an external force in bringing the mass from infinity to that point (in the field).
At a point in infinity, a mass is so far away from the mass generating the gravitational field that the gravitational force due to the mass generating the gravitational field is 0 and the point is defined to have zero gravitational potential energy.
U = -GMm/r (The negative sign affects the magnitude of G.P.E, it does not mean that U is a vector)
For small distances above the surface of the earth, U = mgh.
Explain why gravitational potential energy/gravitational potential is negative in value.
The gravitational potential (or potential energy) at infinity is 0. Since the gravitational force is an attractive force, to bring a mass from infinity to a point in the gravitational field, the direction of the external force is opposite to the direction of displacement of the mass. Hence, the work done by the external force is negative and potential energy goes from 0 to negative. (Recall: Initial energy + work done by external force = final energy / 0 + W = U)
State the relationship between gravitational potential energy and gravitational force.
For a field of force, F = -dU/dr. Hence, the gravitational force acting on a mass m in the presence of another mass M is the negative of the gravitational potential gradient. The magnitude of the force at point r is equal to the gradient of the gravitational potential energy curve at r. The negative sign indicates that the force points in the direction of decreasing potential energy.
Explain how the gravitational potential at a point due to two or more masses can be found.
Adding the individual potentials at that point due to each mass. Points at a fixed distance from a mass form an equipotential surface (of equal interval become further apart as we move away from the mass)
State the relationship between gravitational potential and gravitational field strength.
Gravitational field strength g is the negative of the gravitational potential gradient, g = -dphy/dr. The negative sign indicates that the field strength points in the direction of decreasing potential/points from a higher potential region to a lower potential region.
Define escape velocity. Explain how it can be obtained.
Escape velocity (or escape speed) is the minimum speed needed for the object to just escape from the gravitational influence of a massive body.
At infinity, the object’s GPE is defined to be 0. If the object has sufficient energy to just reach infinity, its KE at infinity is 0. Hence, the Et at infinity = Ep + Ek = 0. From the principle of conservation of energy, the total energy of an object should remain unchanged throughout its motion hence any object with a total energy of zero will be able to reach infinity and stop there. If the object has an energy that is greater than 0, it will be able to go beyond infinity, and completely escape the Earth’s gravitational field.
- GMm/Rearth + 1/2mv^2 = 0
v = square root of 2GM/Rearth = square root of 2gRearth
(to just reach infinity, the escape velocity is independent of the mass of the object)
State the formulae used to calculate the centripetal force acting on an object when it is in a uniform circular motion and the time taken T for it to make one complete circle.
Fc = mac = mrw^2 = mv^2/r where v is the linear velocity and w is the angular velocity.
T = 2pi/w = 2pir/v
Explain why the spring balance indicates a value less than the true weight (value of apparent weight is lower) Fg of an object at the equator and not at the polar regions of the earth.
Pg 17 of notes
Recall: Tension and gravitational force acting on the object as it undergoes uniform circular motion at the equator unlike at the polar regions where the free-fall acceleration will equal the true gravitational field strength g of the Earth at the region. As a result, the free-fall acceleration at the equator will be less than the true gravitational field strength of the Earth at the equator (part of it is used to accelerate the object towards Earth while the other is used to keep it in circular motion)
State Kepler’s third law. Explain how it is derived.
Kepler’s third law states that the square of the period of revolution of planets is directly proportional to the cubes of their mean distances from the sun. (The period and radius of orbit is independent of the mass that is orbiting around the sun)
It is derived by equating gravitational force to centripetal force and converting centripetal force to an equation that includes omega, followed by period.
Derive the formula used to calculate the kinetic energy of a satellite that is in a circular orbit. Then, derive the formula used to calculate the total energy of the satellite. Deduce the sign of the total energy of objects that are bounded and those that will be able to reach infinity and escape.
(Pg 21 of notes)
Recall: The gravitational force on the satellite provides the centripetal force. The total energy of the satellite is equal to the sum of its kinetic and potential energy. For objects that are bounded, their total energy will be negative. However, for objects that are able to reach infinity and escape, their total energy will be equals to or more than 0.
