Circular Motion Chapter 7

Centripetal Acceleration As a car circles a roundabout at constant _____, is there a change in velocity? Yes. Even though magnitude is constant, the _________ is changing. A change in velocity means  v = What does this represent or mean?

According to similar triangles with similar angles, then: similarly So (divide by t) and we get a c = Where v = tangential velocity And a c = which means:

So, anytime an object moves in an arc or full circle, it experiences ___________ _____________. If the object speeds up or slows down during this, then a _____________ _____________ was also experienced, such that there is a ____ __________. at an angle of where  is the angle away from a t towards the radius. acac atat a Nett 

Consider a ball being swung on a rope. If a mass experiences an acceleration, then there must be a Force in the same direction. i.e. F = ma SoF c = F c is called ____________ ______. It is a rotating force that points towards the centre. The __________ ______ on the swinging ball is the ________ in the rope. Horizontal circle – Parallel to the ground

Vertical Circle - Perpendicular to the ground. T = w F R = F C = T = F R = F C = T =

Roller Coasters (Yeah !!) At Pt.A F C = n = At Pt.B F C = n =

Earth has gravity (acceleration) and a Force due to gravity (Weight) because of it’s _____. The Moon’s gravity is _____ that of the Earth’s. Why? Less _____.  Force of gravity is proportional to ______. If the Earth was smaller, but with the same ______, what would happen to gravity? It would __________.  Force of gravity is inversely dependant on the ________. i.e.F  &F  The Earth pulls on the Moon with a Gravitational Force, but The Moon also pulls on the Earth with the same Force. (evidence?)

This Gravitational Force (F G ) was formulated by Isaac Newton (1687) and calculated by: where: G = x N.m 2 /kg 2 (Gravitational constant) M 1 = Object at the centre(e.g. Earth) m 2 = the orbiting object, and(e.g. Moon) r = distance from center to centre of the objects. There is a Gravitational Force acting on any 2 objects that are in proximity to each other simply because of their masses. e.g.’s

Pool Balls Ball 1 is experiencing 2 attractive forces. One from ball 2 ( ) and one from ball 3 ( ). The Resultant Force (F) would cause ball 1 to move in that direction if ______ & _____ were not present.

m 1 = 0.3kg (fixed) m 2 = 0.2kg m 3 = 0.1kg (fixed) r 12 = 0.25m r 23 = 0.50m a 2 = ? F 12 = Gm 1 m 2 r 12 2 F 23 = Gm 2 m 3 r 23 2 F R = F 12 - F 23

Where between m 1 and m 3 should m 2 be placed such that it experiences a zero Nett Force?

So, at the Earth’s surface, a mass (m) experiences a Force = W = and nowF G = Where M 1 is the Earth (M E ) and m & m 2 is the object which can cancel, and r is the radius of the Earth (R E ),  = (6.673x Nm 2 /kg 2 )(5.98x10 24 kg) (6.38x10 6 m) 2 = m/s 2 What about other planets? Use Table 7:3 to solve.

g = 9.80 m/s 2 at the surface but what happens to g as you move away from the Earth? (increases or decreases?) g is _________ ____________ to height. So now, r is the Earth’s radius plus the extra height, i.e.r = so g at any height is: g =

*Note* Remember that Work (W) can equal or Well, now Centripetal Force (F C ) can be the result of any of the following: F C = The last equation can be used to solve for the speed of a satellite (including the Moon) going around the Earth.(also Earth around the Sun.)

Gravitational Potential Energy Remember that P E = = on the Grand scale! Substituting g = And choosing an infinite distance away as the reference level, we get: P E =

Escape Velocity (v esc ) Let’s consider shooting a gun straight up. The bullet reaches a maximum ht & returns. A higher powered rifle will cause the bullet to go _______, but still return. How fast would a bullet (or anything) need to go to escape the Earth’s pull? Initial Energy at the Earth’s surface = Final Energy at some distance far far away! K Ei + P Ei = K Ef + P Ef

½ mv i 2 + [-GM E m/R E ] = ½ mv f 2 + [-GM E m/(R E +h)] Far far away means h =  → P Ef = 0andv f = 0 And we get ½ mv i 2 – GM E m/R E = 0 And finally v i = v esc = Consequences for space travel & Rocket propulsion?

Kepler’s Laws History: 2 A.D.: ________ develops the geocentric model for the Universe. centre) 1543: _______________ develops the ________________ model. centre) But all believed that the planets & stars traveled in _________ orbits. 1600’s: Kepler interprets his mentor’s (Tycho Brahe) data and discovers that the orbits are in fact __________. His labourious calculations led him to his famous 3 laws. (Discuss ellipses & eccentricity)

Kepler’s 1 st Law: All planets move in ___________ orbits around the Sun. The Sun is positioned at one of the ________. The orbits are more circular than elliptical. Kepler’s 2 nd Law: If you draw a line from the Sun to any planet, it will sweep equal _______ in equal ______ intervals. The time to travel A-B is the same for C-D  Areas ASB & CSD are equal. What about velocity during orbit? Time to discuss Summer/Winter?

Kepler’s 3 rd Law: Kepler noticed that there is a proportional relationship between the ____________ of orbit of a planet and its ____________ from the Sun.i.e. Using And We get: T 2 = 4π 2 r 3 = K S r 3 GM S K S = 2.97x s 2 /m 3 Applications?

Some facts to back up Kepler’s 3 rd Law: