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