Circular Motion

Question 1

Radian θ

Answer 1

The angle sub tended from an arc that has a length equal to the radius

• θ = s/r
• unit = radians
  (if s=r then θ = 1 radian)

Question 2

Radians in a full circle

Answer 2

2π

Question 3

Linear Speed (v)

Answer 3

• The distance travelled along the arc with respect to time

- v = s/t

Question 4

Angular Velocity (w)

Answer 4

• The rate of change of the angle with respect to time
• w = θ/t
• unit = rad/sec

Question 5

Periodic Time (T)

Answer 5

• The time taken for one complete revolution of the circle
• T= 2πr/v
• T = 2π/w
• unit = seconds

Question 6

Derivation of linear speed in terms of angular velocity

v = wr

Answer 6

1) θ = s/r 
 divide by t 
2) θ/t = s/rt 
sine  w = θ/t 
3)  w = s/rt
since v = s/t 
4) w = v/r 
5) v = wr 

(θ = angle in rads, r = radius in m)

Question 7

Centripetal Force

Answer 7

• the force directed towards the centre of a circle that is required to maintain circular motion
  (to keep a body moving in a circular path)
• mass x centripetal acceleration
• directed to centre so ⟂ to particle
• F = mv²/r
• F = mrw² (because a = v²/r and a = rw²)
  (e. g tension on string)

Question 8

Force - ⟂

Answer 8

• force applied ⟂ to a particle causes it to turn in a circle
• force directed to centre will be ⟂ to particle

Question 9

Centripetal acceleration

Answer 9

the acceleration towards the centre of a circle that holds a body in a circular orbit

• a = v²/r
• a = rw²
• since particle is constantly moving in a circle
• constantly changing direction
• constantly changing velocity
• constantly accelerating
  (speed always the same)
  (if acceleration is always towards centre there must be a force holding it there)

Question 10

Derivation of centripetal acceleration in terms of angular velocity a = rw²

Answer 10

1) a = v²/r 
since v = rw 
2) a = (rw)²/r 
multiply out brackets
3) a = r²w²/r
divide by r 
4) a = rw²

Question 11

Derivation of formulae for centripetal force

- F = mv²/r

Answer 11

1) F = ma
since a = v²/r
2) F = mv²/r

Question 12

Derivation of formulae for centripetal force

- F = mrw²

Answer 12

1) F = ma
since a = rw²
2) F = mrw²

Question 13

Satelites and Planetary Orbits - V ∝ √mass of the central body

Answer 13

centripetal force (mvr²) = gravitational force

1) mv²/r = GMm/r²
2) mv² = GMm/r
3) v² = GM/r
4) V = √GM/r

• (r+h)
• independent of mass of smaller object
• speed and height linked

Question 14

Derivation of T² = 4π²r²/GM

Answer 14

centripetal force (mrw²) = gravitational force 
1) mrw² = GMm/r²
2) w² = GM/r^3
since T = 2π/w ---- w = 2π/T
3) (2π/T)² = GM/r
4) 4π²/T² = GM/r 
5) T² = 4π²r²/GM

!!!! r = ( r + h )!!!
!!! unit = seconds!!!!

Question 15

Kelper’s 1st law

Answer 15

planet’s move in elliptical orbits around the sun

Question 16

Kelper’s second law

Answer 16

the line joining a planet and the sun sweeps out equal areas during equal intervals of time

Question 17

Kelper’s third law

Answer 17

• T² ∝ r^3

- the square of the period of a planet orbit is proportional to the cube of the radius of it

Question 18

Different planets

Answer 18

• g1/g2 = m1 x r2²/ m2 x r1²
• g = GM/r²
  (G = newton’s gravitational constant)
  (g = gravity)

Question 19

Geostationary Orbits

Answer 19

• stays at same point above earth
• allows it to be found for telecommunication purposes
• height important as determines speed
• must travel at same periodic time as earth (24 hrs)

Must :

• be on equatorial plane
• period = 24 hours (24 x 60 x 60 seconds)
• earth and satellite rotating in the same direction

Question 20

GPS satellites

Answer 20

• not stationary
• don’t stay at same point above earth
• T is not 24 hours