physics / circular.mdon commit [spec] DEs of the dependent variable (a978feb)
   1# Circular forces
   2
   3## Velocity in a circle
   4
   5$$v={{2 \pi r} \over T}$$
   6
   7where $T$ is the period (time for one oscillation)
   8
   9(derived from $v={d \over t}$ where $d=2 \pi r = \pi D$)
  10
  11## Frequency and period
  12
  13$$ f={1 \over T}$$
  14$$ t={1 \over f}$$
  15
  16## Centripetal acceleration & force
  17
  18$$a={{{v^2} \over r}={{4 \pi^2r}\over T^2}}$$
  19
  20where
  21$a$ is centripetal acceleration
  22$v$ is speed
  23$r$ is radius
  24$T$ is period
  25
  26and
  27
  28$v \perp a$
  29
  30We know that $F=ma$, so
  31
  32$$F={{mv^2}\over r}={{4\pi^2rm}\over T^2}$$
  33
  34## Banked track
  35
  36Forces acting:
  37- Weight force $F_g$, vertically down
  38- Normal (reaction) force $R$, perpendicular to slope
  39
  40
  41- Net force  $\Sigma F$ acts towards centre
  42
  43### Calculating the angle
  44
  45$$ tan \theta = {\Sigma F \over F_g} $$
  46
  47where
  48$\Sigma F$ acts towards centre of circle
  49$F_g$ is force by gravity on the moving object
  50
  51In terms of velocity..
  52
  53$$ \tan \theta = {v^2 \over rg}$$
  54
  55$$\therefore \theta = \tan^{-1}({v^2 \over rg})$$
  56
  57$$\therefore v=\sqrt{rg \tan \theta} $$
  58
  59- If $F_N > F_g$, passenger feels heavier
  60- If $F_N < F_g$, passenger feels lighter
  61
  62### Minimum velocity in circle
  63
  64$$ v= \sqrt{gr}$$
  65
  66### Force and acceleration
  67
  68$$\Sigma F = F_N + mg$$
  69
  70## Vertical circular motion
  71
  72## Pulley-mass system
  73
  74$$a={{m_2g}\over{m_1+m_2}}$$
  75
  76where $m_2$ is the suspended mass