Curvilinear Motion [Kinetics]
December 8, 2016
Categorised in: 1st Year Mechanics
Steps for application:
1. Draw F.B.D of particle
2. Apply D’Alembert’s force m x $a_N$ = ${m v^2 }/{ρ}$ in opposite direction of $a_N$ (i.e. away from centre of curvature)
3. Apply Σ$F_N$ = 0
Some important concepts:
(a) When particle is moving along a horizontal circle, mark y axis and normal axis in F.B.D.
(b) When particle is moving along a vertical circle, mark normal and tangential axis in F.B.D.
(c) When a particle attached by a string and whirled in verticle circle
$T_{top} = {mv^2}/{r} – mg$
$T_{bottom} = {mv^2}/{r} + mg$
(d) When an aeroplane flying along a vertical circle, Reaction exerted by seat on the pilot is,
$R_{top} = {mv^2}/{r} – mg$
$R_{bottom} = {mv^2}/{r} + mg$
(e) The normal reactions for a car in following three cases are
$R_1 = mg – ma_N$
$R_2 = mg$
$R_3 = mg + ma_N$
$R_3 > R_2 > R_1$
∙ At top of the path when car lost its contant, R = 0
$mg = ma_N$
$mg = {mv^2}/{r}$
$V_{max} = √{gr}$
∙ When a particle tied with a string and revolves in a vertical cricle
$v_{min} = √{gr}$ to remain in contact with circle
and $v_{bottom} = √{5gr}$
∙When a block is placed on a horizontal turn table at a distance r
The condition for the block not to slip.
$v_{max} = √{μ_s gr}$
∙ When a block is released from top of a cylinder, it will leave the cylinder
When $θ = cos^{-1} (2/3)$
∙ In above case, when it leaves the surface at B, Normal reaction is zero.
∙ When a pendulum is released from horizontal position
At lowest position
$v = √{2gL}$ and T = 3mg
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