Circular Motion I

Circular Motion I

Circular Motion: Body moving in circle at constant speed will accelerate towards the centre

Dimensionally: must be combination of v and r that gives dimension of accn.

[r] = L [v] = LT-1 [a] = LT-2 so that only possibility is

a = v2 = (LT-1)2 r L = L2T-2 = LT-2 L

Centripetal Acceleration

This is called centripetal accn. It is NOT centrifugal accn. (which is the apparent accn. that a body feels in a rotating frame of reference)

Centripetal force is force required for centrip. accn. It does not exist as a force in its own right: it has to be supplied by another force: e.g.

• String
• Tension
• Friction
• Gravitation
• Magnetism

A kid whirls a stone around which is tied to a piece of string of length 50 cm. The string has a breaking strain of 20 N, and the stone weighs 400 gm. How fast is the stone going when it the string breaks? ignore the fact that the rope gets pulled down by the mass include this. Does it make a difference?

A car rounding a corner on a flat road relies on friction to provide the centripetal force. e.g. speed = 30 ms-1, mass = 1500 kg, R = 300m, what must μ be?

Note that friction provides an acceleration here: the speed does not change

On a banked curve, the friction can be replaced by the horizontal component of the reaction e.g.for the car in the last example, what would the slope need to be so that there was no sideways force on the wheels?

How about a race track?

On race tracks, curves are usually banked with the steepest part of the bank at the top. This is because If a car starts sliding up the bank, it will get steeper and so the centripetal force will increase It is to allow the car to go faster near the bottom of the bank. It is to make the cars collide with the retaining wall at the top of the bank as often as possible

Newton's Discovery of the law of Universal Gravitation

Kepler's laws : founded on observations by Tycho Brahe

• Planets move in ellipses, with one focus at the sun

• A vector drawn from the planet to the sun will sweep out equal areas in equal times

• The period (P) and the semi-major axis (r) are related by

  P2= constant r3

v = 2πr
    P     

 F = m v2
      r

F = mg

• How does the moon stay up?

• By falling!
• faster
• and faster
• So how long would the month be in this case?
• R = 400000 km and this gives.....
• P = .....

Scratch one theory

Need extra ingredient of Kepler's laws

What kind of gravitational force can give

P2= constant r3

Obviously	v = 2πr P
and centripetal force	F = m v2 r
So	F = k R2

But this only refers to Sun: need to find law of same form, which depends on the mass of the planet, and the mass of the sun.

1. MLT
2. MLT^-2
3. LT^-2
4. MLT^-1

Given that the dimensions of force are MLT^-2, what are the dimensions of G?

1. MLT^-2
2. M^-1L³T^-2
3. ML³T^-2
4. M^-1L^-3T^-1

Law of universal gravitation

Applied to earth-moon: need to know mass of earth (M) and G (not mass of the moon, since it will cancel out),

but we do know g at earth's surface

g = G M R02

where R₀ is the radius of the earth, and all we really need is the product GM

Hence at the moon

F = mg (R0/R)2

We can then equate grav. force to centrip. force to give......

• \color{red}{ P = 2\pi \sqrt {\frac{{R^3 }}{{gR_0^3 }}} }
• = ??
• = 27.4 days (!!!!!!!!!!!)
• Note the logical flow
• This was the first time that laws deduced on the Earth were seen to apply outside!

Gravitational force between any two bodies, masses M1 and M2 separated by distance R is given by Newton's Law of universal gravitation

\color{red}{ F = \frac{{Gm_1 m_2 }}{{R^2 }}}

Note that this imples that grav. force gets weaker as we move away from the earth

G=6.67x10^-11 N m² kg^-2 is a universal constant Now we need to do some WORK