Gravitational and Electric Potential Energy, Study notes of Physics

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TOPIC 10

FIELDS

DESCRIBING FIELDS

● we can think of fields as ‘force fields’ present in space ⇒ particles which are sensitive to the particular field feel the associated force when placed in the field

GRAVITATIONAL FIELD: a space where a small test mass experineces a force due to

another mass - anything with mass

ELECTRIC FIELD: a space where a small +ve test charge experience a force per unit

charge all charged particles ⇒ both fields have a source (as opposed to magnetic field) ⇒ fields take on a different interpretation in the quantum world

GRAVITATIONAL FIELD

NEWTON’S LAW OF GRAVITATION: 𝐹 =

𝐺𝑀 1 𝑀 2 𝑟^2

● Any body with mass M produces a GRAVITATIONAL FIELD around it with the

magnitude (field strength)

𝐺𝑀 𝑟^2 ● the large mass M is the SOURCE of gravitational field ● the field lines emanate from the source radially and are directed

TOWARDS the source (at the centre)

⇒ if a small mass m planed near (subtlety!*), this mass feels an

ATTRACTIVE force with the magnitude

𝐺𝑀𝑚 𝑟 2

* the field strength decays as ∼ so it never quite reaches zero unless you are

1 𝑟^2 infinitely far from the source ***** note that the large mass also feels a force with the same magnitude directed towards the small mass (Newton’s 3rd Law)

GRAVITATIONAL POTENTIAL ENERGY (GPE)

● consider the above scenario of the 2 masses M & m - they both feel an attractive force towards each other ● GPE is stored in the field = an energy shared between the 2 masses ❖ when we talk about the GPE of Earth, we use formula:

𝑝

2 things to note: ● the Earth is approximately flat very close to its surface (we don’t feel curvature in our daily lives) ⇒ the field is approximately uniform meaning g = 9.8ms¯² is approximately constant!

● we only talk about the CHANGE in height and the CHANGE in GPE ⇒ what do we

take as the reference ‘height’ when defining GPE? - SPATIAL INFINITY

GPE OF 2 BODIES: the work that was done in bringing the bodies to their present

position form when they were infinitely apart Apart from the notion of infinity in space, we also neglect the change in the KE involved (very small constant speed) Recall: 𝑊 = 𝐹𝑠⇒ for a constant force F ● for a spatially varying force ( eg: gravitational force), the definition of work is:

● the work done in bringing the small mass m from INFINITY to a DISTANCE r from

the large mass M is:

∞ 𝑟

𝐺𝑀𝑚 𝑟'^

'

= [−

𝐺𝑀𝑚 𝑟'^

]

∞ 𝑟

𝐺𝑀𝑚 𝑟 ● the gravitational potential energy stored in such a situation is:

−𝐺𝑀𝑚 𝑟 the work done by the external agent to bring in the mass m from infinity ***** force of gravity is attractive ● to separate the masses infinitely, we need to do work against the gravitational field:

𝐺𝑀𝑚 𝑟 EG: to move away from the Earth surface, we need to put in energy ( eg: jumping)

FIELD STRENGTH: force per unit mass (gravitational) / force per unit charge (electric)

GRAVITATIONAL POTENTIAL at a point P in a gravitational field is the work done per unit

mass in bringing a small point mass m from infinity to the point P ⇒ always -ve ⇒ at infinity = 0 ***** to calculate the gravitational potential due to multiple masses: simply add up the gravitational potential due to the individual masses

● since work is a scalar quantity, the gravitational potential is also a SCALAR

quantity (c.f. forces & field strengths are vector quantities)

𝐺𝑀 𝑟 ● neglecting KE, work done to move from point A to B in a gravitational field is the change in GPE

𝐵

𝐴

*** PATH INDEPENDENT** quantity

EQUIPOTENTIAL SURFACES

ELECTRIC POTENTIAL: 𝑉

𝑒

𝑘𝑄 𝑟

GRAVITATIONAL POTENTIAL: 𝑉

𝑔

𝐺𝑀 𝑟 ● the points with the same distance r from the source (point charge / mass) have

the SAME POTENTIAL

⇒ they lie on EQUIPOTENTIAL SURFACES

EQUIPOTENTIAL SURFACE: a surface consisting of all the points with

the same potential

⇒ for a point charge / mass, these are CONCENTRIC CIRCLES

𝑑𝑉𝑒

𝑑𝑟 𝑔 =^ −^

𝑑𝑉𝑔 𝑑𝑟 (electric) (gravitational)

* field strength is always given by the gradient of the graph of potential

POTENTIAL GRADIENT:

∆𝑉 ∆𝑟 ⇒ where ∆V = change ingravitaitonal potential between 2 points & ∆r = distance between the 2 points ⇒ slope of a graph which plots the graviational potential against the distance from the mass

∆𝑉

∆𝑟 =^

𝐺𝑀 𝑟^2

FIELD LINES & EQUIPOTENTIAL SURFACES

● density of field lines is proportional to the field strength ● field lines & equipotential surfaces are perpendicular to each other ● can be understood in terms of work done: if we move a mass / charge along the

equipotential surface, the work done is 0 ⇒ ∆V = 0

● field lines (which indicate the direction of force) are perpendicular to the equipotential ⇒ force is perpendicular to direction of motion - no work done EG: field lines - red / equipotential line - black 2 equal charges

2 unequal charges

PARALLEL PLATES

● the electric field between 2 parallel plates is UNIFORM (denoted by straight lines)

● the equipotential surfaces are therefore also straight lines, just perpendicular to the field of lines ● if the potential difference between the plates is V , then the electric field strength is

𝑉 𝑑

ORBITAL MOTION

● assume a satellite with mass m orbiting around a large planet with mass M , such that M ≫ m ● due to the mass difference, we can assume that the planet is not moving ⇒ the forces are still equal and opposite! ⇒ the accelerations are different due to the mass difference ● the total energy of this system (satellite + planet) is total energy = KE of satellite + PE shared by planet and satellite

𝑇

1

2

𝐺𝑀𝑚 𝑟

KEPLER’S 3RD LAW: the square of a planet’s period is proportional to the cube of its

mean disrance from the Sun

2

3 ● assuming that a planet’s orbit is circular (which is not exactly correct but is a good approximation in most cases), then the mean distance from the SUn is a constant ?> the radius ● F is the force of gravity on the planet + F is the centripetal force

● If the orbit is circular, the planet’s speed is constant & 𝑣 =

2π𝑅 𝑇 𝐺𝑀𝑚 𝑅^2

𝑚𝑣 2

𝑅 =^

𝑚 2π𝑅/𝑇[ ] 2 𝑅

The possible paths of the satellite are related to the sign of 𝐸𝑇! 𝐸 : bound - orbits the planets (circular / elliptical orbit) 𝑇

𝐸𝑇 = 0 : free - shoots off to infinity & stops there (parabolic path) 𝐸𝑇 > 0 : free, shoots off and keeps moving (hyperbolic path)

ESCAPE VELOCITY: minimum speed required for the satellite to “escape” the planet -

never return / stay in orbit around ⇒ an object launched at / above its escape speed will not return to the planet due to gravity

𝑇

1

2

𝐺𝑀𝑚

***** where R is the radius of the planet

𝑒𝑠𝑐

2𝐺𝑀 𝑅