Linear motion

• Average velocity -> v = Δd/Δt

  
  

  Average acceleration -> a = Δv/Δt
• Kinematic equations (not average acceleration or velocity)

  • vf = vi + a x t
  • vf$^2$ = vi$^2$ + 2a x d
  • d = (vi + vf/2) × t
  • d = vi x t + 1/2 x a x t$^2$
• Circular motion

  This is when objects travel in circles.

  
  

  Circumference = 2πr

  
  

  To find how far an object has travelled: Circumference x number of rations of the circle by an object

  
  

  If the object has done partial rotations, distance travelled: d = rθ

  d is distance(m), r is radius(m), θ is angle travelled
• Speed

  Angular speed is the number of radians around a circle an object travels in each second. 

  
  

  For circular motion, frequency is the number of times an object goes round the circle every second

  
  

  Angular speed: ω = 2πf

  connecting angular speed with linear: v=rω,

  linear speed(v), angular speed(ω)
• Centripetal force


  A force toward the centre of the circle is what causes circular motion. The inward force is constantly pulling the object in from travelling in a straight line, forcing it to move in a circle. This inward force is called centripetal force.

  
  

  If that force suddenly stopped existing, the object would fly off at a tangent to the circle. (Newtons first law - when no forces acting, object will travel at constant velocity, same speed & direction)
• Centripetal force

  In order for an object to travel in a circle, there must be a force acting on it that acts toward the centre of the circle. This force is called the centripetal force(Fc) and is provided by tension or support or friction or gravity.
• Centripetal force(Fc) need to be the right size, it depends on object's speed(v) and radius(r).

  
  

  -> faster speeds need bigger forces because there are faster changes in direction.

  -> bigger the radius smaller the inward forces because big circles mean slower turns.

  -> heavier objects need bigger forces

  
  

  Equation: Fc = m x v$^2$/r
• Centripetal force isn't constant


  Fc changes -> mass fixed, so velocity and/or radius needs to change. If Fc increases, v and/or r decreases (vice versa).

  sometimes radius fixed, so velocity has to change so -> Fc $∝$v$^2$
• example diagram

  
• example working

  
• Gravity


  Gravitational force, Fg, pulls all objects together.

  It acts between objects causing equal and opposite forces (Newton's third law), Force is stronger when objects are closer or have more mass.
• Fg = GMm/r$^2$

  
• Can use Fg = GMm/r$^2$ to find acceleration due to gravity, g (also gravitational field strength)

  F=ma -> A = F/m, for gravity trade terms

  => g =fg/m

  • This is how fast an object with mass m would accelerate towards object with mass M given no other net forces.
  • Changes over time, When objects get closer, r decreases, meaning Fg decreases.
• Small object orbiting large object -> gravity provides inward centripetal force that causes circular motion. Fc = Fg. (r will be distance between centres of mass, add height of small object to large object radius)
• The gravitational force from Earth (or any planet) acts as a centripetal force on its satellites. This means that Fg = Fc or GMm/r$^2$= mv$^2$/r, which can be solved for M, m, v or r.
• • A geostationary or geosynchronous orbit is where a satellite stays right above the same spot on Earth. To do this, it has to go at exactly the right speed, so that it does one orbit of the Earth per day (to keep up with the Earth’s rotation).
  • Relate d to t, distance the satellite travels in a period is the circumference of its orbit

  so v = distance/time = 2πr/t
• Fc = m x v$^2$ -> v = √Fc x r/ m
• v = d/t -> t=d/v
• Question part 1

  Earth has a mass of 5.972 ×10$^2$$^4$kg. The Moon has a mass of 7.348 × 10$^2$$^2$kg. The moon orbits Earth at a distance of approximately 385,000 km. How long is the Moon’s orbit?

  -> first find Fg, bc its orbiting Fc = Fg use that to find speed

  he Moon’s orbit?

  -> first find Fg, bc its orbiting Fc = Fg use that to find speed

  
• Question part 2

  Now we have speed find out time to orbit.