First slide

Simple hormonic motion

Question

A sphere of radius r is placed on a concave surface of radius of curvature 5 r, which itself is placed on a horizontal table as shown in fig. (3). When the sphere
is displaced from the equilibrium position it describes S.H.M. of period

Moderate

A

$2 π (r / g)^{1 / 2}$
B

$4 π (r / g)^{1 / 2}$
C

$2 π (5 r / g)^{1 / 2}$
D

$2 π (6 r / g)^{1 / 2}$

Solution

Here, restoring force = - M g sin $θ$ Considering the displaced position, we have
$\sin θ = X / (5 r - r) = X / 4 r ∴ Ma = - Mg (x / 4 r) a = - (g / 4 r) X so T = 2 π \sqrt{(\frac{4 r}{g})} = 4 π \sqrt{(\frac{r}{g})}$

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