Circular Motion Concepts Page - 5

Example

Rigid bodies rotating in vertical circle

Example:  When a bucket containing water is rotated fast in a vertical circle of
radius R$R$,  When the water in the bucket doesn't spill?

Solution:
For water not to spill,    Mar=Mg$M{a}_r=Mg$
Mv2R=Mg⟹v=gR−−−√${\displaystyle \frac{M{v}^2}{R}}=Mg⟹v=\sqrt{gR}$
Now w=vR=gR−−√$w={\displaystyle \frac{v}{R}}=\sqrt{{\displaystyle \frac{g}{R}}}$w=2πν⟹ν=g4π2R−−−−−√$w=2\pi \nu ⟹\nu =\sqrt{{\displaystyle \frac{g}{4{\pi }^{2}R}}}$  s−1$s^{-1}$ (OR  r.p.s)ν=g4π2R−−−−−√×60$\nu =\sqrt{{\displaystyle \frac{g}{4{\pi }^{2}R}}}\times 60$  r.p.m$r.p.m$Thus  ν=900gπ2R−−−−−√$\nu =\sqrt{{\displaystyle \frac{900g}{{\pi }^{2}R}}}$  r.p.m

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