First slide

Simple hormonic motion

Question

A solid cylinder of mass M and radius R is connected to spring of force constant k as shown in fig. Cylinder rolls without slipping on the horizontal surface. Its time period of oscillation is

Difficult

A

$T = 2 π \sqrt{\frac{2 M}{3 K}}$
B

$T = 2 π \sqrt{\frac{2 M}{K}}$
C

$T = 2 π \sqrt{\frac{M}{3 K}}$
D

$T = 2 π \sqrt{\frac{3 M}{2 K}}$

Solution

Total energy $\frac{1}{2} {Mu}^{2} (1 + \frac{K^{2}}{R^{2}}) + \frac{1}{2} {Kx}^{2} = constant$
$\frac{1}{2} {Mu}^{2} (1 + \frac{1}{2}) + \frac{1}{2} {Kx}^{2} = constant$
$Mu \frac{du}{dt} (\frac{3}{2}) + Kx \frac{dx}{dt} = 0$
$Mu (\frac{du}{dt}) (\frac{3}{2}) + Kxu = 0$
$\frac{3 M}{2} a + Kx = 0$
$w = \sqrt{\frac{2 k}{3 M}} T = \frac{2 π}{w} = 2 π \sqrt{\frac{3 M}{2 K}}$

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