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Physics

1.4 Gravitation

Carry out calculations involving Newton’s universal law of gravitation.

Newton’s universal law of gravitation states that there is a force of attraction between any two massive particles in the universe. The magnitude of the force is directly proportional to the product of the masses & inversely proportional to the square

of the distance between the masses.

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Carry out calculations involving Newton’s universal law of gravitation.

F =Gm1m2

r2

𝐹 = 𝑓𝑜𝑟𝑐𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝐺 = 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 m1 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑜𝑏𝑗𝑒𝑐𝑡 m2 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑠𝑒𝑐𝑜𝑛𝑑 𝑜𝑏𝑗𝑒𝑐𝑡 r = distance between the objects

Carry out calculations involving Newton’s universal law of gravitation.

Example:

Alvin and Anne have a mass of 65 kg and 50 kg respectively. Determine the gravitational force on each of them if there are separated by a distance of 10 cm.

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Define gravitational field strength.

A gravitational field is a region where an object with mass experiences gravitational force.

Every object has a gravitational field associated with it due to its mass.

Define gravitational field strength.

When another object with mass (m) enters this gravitational field, the new mass will experience a force.

The term gravitational force is used because the force experienced is due to gravity (i.e. mass).

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Define gravitational field strength.

Gravitational field is a vector quantity with units 𝑁 𝑘𝑔−1.

Gravitational field strength refers to the magnitude of the gravitational field at a point.

The gravitational field strength on earth’s surface is 9.81 𝑁 𝑘𝑔−1.

The amount of 9.81 𝑁 𝑘𝑔−1 tells us that:

a) A 1 kg mass will experience a force of 9.81 N on earth’s surface.

b) A 1 kg mass will accelerate at 9.81 𝑚 𝑠−2 on earth’s surface.

Define gravitational field strength.

Gravitational field strength (at a point) is defined as the gravitational force per unit mass (at that point).

g =F

m

𝑔 = 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ 𝐹 = 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 m = 𝑚𝑎𝑠𝑠

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Sketch gravitational field lines for an isolated point mass and for two point masses.

The gravitational field surrounding a mass can be represented by lines (field lines).

The arrow head represents the direction.

The number of lines per area represents the magnitude. So, closer lines means stronger field.

Sketch gravitational field lines for an isolated point mass and for two point masses.

The lines are always perpendicular to the surface of the mass.

The lines will never intersect.

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State that the gravitational potential at a point in a gravitational field is the work done by external forces in bringing unit mass from infinity to

that point.

Gravitational potential at a point is defined as the work done by an external force in bringing unit mass from infinity to that point.

It also refers to the potential energy per unit mass at a point in a gravitational field.

State that the gravitational potential at a point in a gravitational field is the work done by external forces in bringing unit mass from infinity to

that point.

Gravitational potential is a scalar quantity with units 𝐽 𝑘𝑔−1

It also refers to the potential energy per unit mass at a point in a gravitational field.

𝑉 = −𝐺𝑀

𝑟

𝑉 = 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐺 = 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑚 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑖𝑠 𝑛𝑒𝑎𝑟 𝑟 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡 𝑓𝑟𝑜𝑚 𝑜𝑏𝑗𝑒𝑐𝑡

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State that the zero of gravitational potential is taken to be at infinity.

At infinity, the gravitational potential is zero.

This makes the gravitational potential values to be always negative because the work done (by the external force) in bringing a unit mass from infinity to any point in the field is negative work*.

*Positive work occurs when we use a force to lift a book against gravitational force. Negative work is when we put the book down following gravitational force.

Carry out calculations involving the gravitational potential energy of a mass in gravitational field.

Gravitational potential energy,

𝐸 = −𝐺𝑀𝑚

𝑟

V = gravitational potential G = gravitational constant m = mass of object the point is near r = distance of point from object

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Carry out calculations involving the gravitational potential energy of a mass in gravitational field.

Example:

A 300 kg satellite orbits at a height of 35 km above the earth’s surface. Calculate:

1) The gravitational potential at this height.

2) The gravitational potential energy of the satellite.

Explain what is meant by a conservative field.

A conservative field is a force field whereby the work done to move an object from point A to B is the same irrespective of the path taken in moving the object.

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State that a gravitational field is a conservative field.

In a gravitational field, the amount of work done to move a mass from point A to point B is the same regardless of the path taken.

So, this makes the gravitational field a conservative field.

Explain the term ‘escape velocity’.

‘Escape velocity’ is the minimum velocity that an object needs to break free of a planet gravitational field to without further propulsion.

It is the speed at which the sum of an object's kinetic energy and its gravitational potential energy is equal to zero.

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Derive the expression 𝑣 =2𝐺𝑀

𝑟 for the escape velocity.

Applying the principle of conservation of energy,

𝐸𝑘 + 𝐸𝑝 = 0

1

2𝑚𝑣2 + −

𝐺𝑀𝑚

𝑟= 0

𝑣 =2𝐺𝑀

𝑟

v = escape velocity G = universal gravitational constant M = mass of the planet r = distance of the object from planet

Derive the expression 𝑣 =2𝐺𝑀

𝑟 for the escape velocity.

Example:

Calculate the required escape velocity to free a rocket from earth’s gravitational field. The earth’s mass is 5.97 × 1024 𝑘𝑔 and it’s radius is 6371 km.

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State that the motion of photons is affected by gravitational fields.

Even the motion of a photon is affected by gravitational fields.

Light that travel close to a extremely dense object can be attracted by its huge gravitational field and move in a curved path instead of a straight line.

State that, within a certain distance from a sufficiently dense object, the escape velocity is greater than c, hence nothing can escape from

such an object – a black hole.

A region of space having a gravitational field so intense that nothing can escape it is called a black hole.

Because the a black hole is so dense and it’s gravitational field strength is so huge, the required escape velocity is greater than 3 × 108 𝑚 𝑠−1.

This means that the speed of light is not enough to escape from a black hole.

So, photons that are traveling too close to a black hole will be unable to escape.

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Carry out calculations involving orbital speed, period of rotation and radius of orbit of satellites.

The orbit of a satellite around a planet is maintained by the gravitational force of the planet.

So, gravitational force is acting as the centripetal force.

𝐺𝑀𝑚

𝑟2 =𝑚𝑣2

𝑟

𝑣 =𝐺𝑀

𝑟

𝑣 = 𝑜𝑟𝑏𝑖𝑡𝑎𝑙 𝑠𝑝𝑒𝑒𝑑 𝐺 = 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑀 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑝𝑙𝑎𝑛𝑒𝑡 𝑟 = 𝑜𝑟𝑏𝑖𝑡𝑎𝑙 𝑟𝑎𝑑𝑖𝑢𝑠

Carry out calculations involving orbital speed, period of rotation and radius of orbit of satellites.

𝑣 =𝐺𝑀

𝑟

2𝜋𝑟

𝑇=

𝐺𝑀

𝑟

𝑇 =4𝜋2𝑟3

𝐺𝑀

𝑇 = 𝑝𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛/𝑜𝑟𝑏𝑖𝑡 𝐺 = 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑀 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑝𝑙𝑎𝑛𝑒𝑡 𝑟 = 𝑜𝑟𝑏𝑖𝑡𝑎𝑙 𝑟𝑎𝑑𝑖𝑢𝑠

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Carry out calculations involving orbital speed, period of rotation and radius of orbit of satellites.

Example:

A geostationary satellites orbits with a period of 24 h.

Calculate:

1) The height of the satellite from the earth’s surface.

2) The satellite’s orbital velocity.