Question

A cylinder of radius R, length l and density
$\gamma$
floats upright in a fluid of density
$\gamma o$
. The cylinder is given a gentle downward push as a result of which there is a vertical displacement of size x; it is then released; the time period of resulting undamped oscillations is
(A)
$2\pi \sqrt{ \frac{ \gamma o l}{ \gamma g} }$
(B)
$2\pi \sqrt{ \frac{ \gamma g}{ \gamma ol} }$
(C)
$2\pi \sqrt{ \frac{ \gamma l}{ \gamma og} }$
(D)
$2\pi \sqrt{ \frac{ \gamma og}{ \gamma a} }$


​

Answer 1

Answer:

Correct option is

B

2π

ρg

ρ

0

l

At equilibrium the weight of the body and buoyant force balance each other.

When body is displace down by x, then the upward force is due to increase in buoyant force due to more displacement of liquid.

F=−Vρg=−ρAxg −ve sign is because displacement and force are in opposite direction

Acceleration=a=

M

−rhoAxg

=

ρ

o

AL

−ρAg

x=−ω

2

x

ω=

ρ

o

L

ρg

Time period =T=

ω

2π

=2π

ρg

ρ

o

L