Solution:

We know that time period of simple pendulum is,

$T=2πgl =2sec−−−−−−−−−−(i)$

Now it makes angle $36_{0}$ when car is moving in a circle. Therefore

component of acceleration due to net force acting on the string will be

$=g_{2}+a_{2} $

New Time period, $T_{′}$, will be

$T_{′}=2πg_{2}+a_{2} l $

$T_{′}=2πg1+g_{2}a_{2} l −−−−−−−−−−−(ii)$

Also,

$tanθ=a/g$ -------- $(iii)$

Solving $i,ii$ and $iii$, we get

$T_{′}=2cosθ $

$=20.81 =1.8sec$

$T=2πI/mgL_{c} $ where $L_{c}$ is length of pendulum in cm.

Where $l$ tends to infinity.

The time taken by the wave for one complete oscillation for the density or pressure of the medium is called the time period. It is measured in seconds.

e.g. The time period of the minute hand is 60 minutes.

A common hydrometer has $1$ and $0.8$ specific gravity marks $4$ cm apart. Calculate the time period of vertical oscillations when it floats in water. Neglect resistance of water.

lets say $l$ is the length of the hydrometer when it is dipped in water and $A$ is the cross section area of the hydrometer, then $l+4$cm length of the hydrometer will be dipped when it is placed in liquid of specific gravity 0.8. So we have

weight of hydrometer=water displaced = liquid (0.8 s.g.) displaced

$⇒l×A×1×g=(l+4)×A×0.8×g⇒l=(l+4)0.8⇒l=16cm=.16m$

And time period of a floating cylinder (hydrometer )is given by $T=2πgl $ where $l$ is the length of the cylinder (hydrometer) dipped in water.

$⇒T=2π9.80.16 =0.8s$

where

$T:$ Time period, $l:$ length, $g:$ acceleration due to gravity

1. The time period of oscillation is directly proportional to to the square root of its effective length.

2. The time period of oscillation is inversely proportional to the square root of acceleration due to gravity.

3. The time period of oscillation does not depend on the mass or material of the body suspended.

4. The time period of oscillation does not depend on the extent of swing on either side.

$T=2πgL $

hence,

$T∝L_{21}$

The graph between time period and length becomes parabola as iterpreted in figure.