Centripetal Acceleration

- In circular motion, an object moves along a perfectly circular path
- Its speed remains the same
- The direction it travels in is constantly changing
- Its velocity is therefore constantly changing
- It is constantly accelerating towards the centre of the circle
- The direction it’s moving in is always at 90° to the direction to the centre
- An object with a force or acceleration acting on it perpendicular to its direction of travel will undergo circular motion
- The acceleration towards the centre is called the centripetal acceleration
- If the centripetal force / acceleration is removed, the orbiting object continues in the direction it was travelling that instant – a tangent to the circle
- The centripetal force can be provided by tension in a piece of string, or by gravity, for example
- (I’m sure you’re aware that the centrifugal force doesn’t exist – you can feel it because of your momentum straining against the centripetal force, but to the outside observer, there’s nothing there)
- No work is done, because Work = Force × Distance Moved in Direction of Force, and the distance moved in the direction of the force is zero, because the orbiting object doesn’t actually get any closer to the centre, so the force is multiplied by zero
- No work done means no energy transferred – objects in circular motion do not use up energy (well, unless they collide with something such as a bunch of air molecules, but we’re ignoring air resistance – especially when in space)
- Now that we’ve got that out of the way…

Set Calculators to Radians

- l = r θ

Where l is the length of an arc (part of the circumference of a circle), r is the radius of the circle and θ is the angle of the sector the arc is from

Think of the sector as a slice of pizza, with r the length of one of the straight, cut sides and l the length of the curved crust

This equation ONLY WORKS IF θ IS IN RADIANS - ω = Δθ/Δt = v/r

ω is the angular velocity, which is the rate of change of angular displacement. It’s measured in radians per second so can be calculated by dividing the angle the object has orbited by the time it took

ω is equivalent to the linear (proper) velocity, v, divided by the radius of the circle - a = vω = rω
^{2}= v^{2}/r

The angular acceleration, a (with the same symbol as regular acceleration) is the angular velocity multiplied by the linear velocity, for some reason

Anyway, substituting Equation 2 into it gives a = v^{2}/r, which is useful - F = mv
^{2}/r

This comes from substituting the acceleration above into F = ma

It’s an equation for the centripetal force in any circular motion - v = (2πr) / T

The linear velocity (at any time, in no particular direction… so the speed?) of the orbiting object is the distance it covers in one orbit divided by the time it takes to cover that distance, i.e. the circumference (2πr) divided by the period (T). This’ll be in metres per second if the circumference is in metres and period is in seconds - F = -(GMm) / r
^{2 }F is the gravitational force between two masses (although gravity isn’t a force – weight is), in Newtons, G is the gravitational constant, M is the mass of the larger object (at the centre of the circle), m is the mass of the smaller object (the one that’s orbiting the larger one) and r is the distance between the two objects

This equation is just here so that I can number it in order to make a point later in the post. I may write a proper post about the gravitational field equations, although that’s looking a bit unlikely since the exam is… about 90 minutes from now, at the time of writing and I urgently need to do last-minute revision on relativistic and Doppler stuff…

Planets’ Orbits

- This physics course treats planets’ orbital paths as perfect circles, unless otherwise specified
- Planets orbiting the sun or satellites orbiting planets are modelled using circular motion
- The planets / satellites themselves are treated as point masses – they have a centre of mass but their dimensions are ignored
- Remember that the surface of a planet is still some distance away from where the point mass is – you may have to add the radius of the planet to the distance above the surface if you need the distance from the centre of the planet
- Normally this won’t be necessary because you’ll be given the correct numbers in the question, but it could be used as an attempt to catch people out
- Experimental Tip: To get a precise measurement of a planet’s orbital period, the time it takes to complete a lot of orbits can be measured and divided by the number of orbits (effectively averaging lots of repeats). Simply measuring the time for one orbit would mean a greater uncertainty
- T
^{2}= (4π^{2}r^{3})/GM is known as Kepler’s Third Law - Since gravity provides the centripetal force when an object orbits another object in space, the gravitational force is equal to the centripetal force. You can equate the letter ‘F’s in equations 4 and 6 above, then substitute equation 5 in to eliminate v. If you rearrange the resulting expression, you should get Kepler’s Third Law

An Elliptical Orbit

- An object in a circular orbit stays at the same distance from the thing it’s orbiting at all times, and its speed is constant
- As mentioned earlier, no work is done by the centripetal force, so no energy is transferred
- An object in an elliptical orbit, however, does get closer to and further away from the thing it’s orbiting, at different points in its orbit
- This results in it speeding up and slowing down
- The distance it moves in the direction of the centripetal force changes, so work is done and energy is transferred
- The energy is converted between gravitational potential energy and kinetic energy
- The total amount of energy does not change, so the kinetic energy added to the potential energy would produce the same result at any point in the orbit
- By convention, gravitational potential energy is negative (as would be discussed in a post on non-uniform gravitational fields and potential wells, if I had time to write one)

Data, Formulae and Relationships Booklet

- Equation 1 above is given (in the Mathematical Constants and Equations section at the bottom of the first page), as arc = rθ, although you’ll have to remember to USE RADIANS
- It also has formulae for the area and circumference of a circle for a given radius
- The formula booklet doesn’t bother with angular velocity – I’m not sure whether it’s part of the course or just used to explain equation 4
- a = v
^{2}/r and F = mv^{2}/r are in the formula booklet - Equation 5 doesn’t seem to be in the formula booklet, but it’s basically speed = distance divided by time (which you surely must remember!) with the circumference of a circle (given at the bottom of the first page of the booklet) as the distance
- Equation 6 is in the formula booklet, in the Field and Potential section
- Kepler’s Third Law doesn’t seem to be in the booklet, but I did explain how to derive it (and it only ever seems to be used in “show that” type questions)

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