MCQ
The bob of a simple pendulum is displaced from its equilibrium position $O$ to a position $Q$ which is at height h above $O$ and the bob is then released. Assuming the mass of the bob to be $m$ and time period of oscillations to be $2.0\, sec$, the tension in the string when the bob passes through $O$ is
  • $m\,(g + \pi \sqrt {2g\,h} )$
  • B
    $m\,(g + \sqrt {{\pi ^2}g\,h} )$
  • C
    $m\,\left( {g + \sqrt {\frac{{{\pi ^2}}}{2}g\,h} } \right)$
  • D
    $m\,\left( {g + \sqrt {\frac{{{\pi ^2}}}{3}g\,h} } \right)$

Answer

Correct option: A.
$m\,(g + \pi \sqrt {2g\,h} )$
a
(a) Tension in the string when bob passes through lowest point 

$T = mg + \frac{{m{v^2}}}{r} = mg + mv\omega $ ( $v = r\omega$) 

putting $v = \sqrt {2gh} $ and $\omega= \frac{{2\pi }}{T} = \frac{{2\pi }}{2} = \pi $

we get $T = m\;(g + \pi \sqrt {2gh} )$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Similar questions

How much work is required to making a soap bubble solution whose radius is r?
An ideal diatomic gas is heated at constant pressure. The ratio of the work done to the heat supplied is
In case of a forced vibration, the resonance wave becomes very sharp when the
Two solid spheres $A$ and $B$ of equal volumes but of different densities $d_A$ and $d_B$ are connected by a string. They are fully immersed in a fluid of density $d_F$. They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

$(A)$ $d_Ad_F$  $(B)$ $d_B > d_F$ $(C)$ $d_A>d_F$ $(D)$ $d_A+d_B=2 d_F$

A vertical spring with force constant $k$ is fixed on a table. A ball of mass $m$ at a height $h$ above the free upper end of the spring falls vertically on the spring so that the spring is compressed by a distance $d.$ The net work done in the process is
A ideal monoatomic gas is carried around the cycle $ABCDA$ as shown in the fig. The efficiency of the gas cycle is
There is some liquid in a closed bottle. The amount of liquid is continuously decreasing. The vapour in the remaining part:
A barometer is constructed using a liquid (density $\left.=760 \;kg / m ^{3}\right) .$ What would be the height  (In $m$) of the liquid column, when a mercury barometer reads $76 \;cm ?$ (density of mercury $\left.=13600 \;kg / m ^{3}\right)$
A cubical solid aluminium (bulk modulus $=-V \frac{ dP }{ dV }=70 GPa$ ) block has an edge length of $1 m$ on the surface of the earth. It is kept on the floor of a $5 km$ deep ocean. Taking the average density of water and the acceleration due to gravity to be $10^3 kg m ^{-3}$ and $10 ms ^{-2}$, respectively, the change in the edge length of the block in $mm$ is . . . . .
A mass $m$ is suspended from the two coupled springs connected in series. The force constant for springs are ${K_1}$ and ${K_2}$. The time period of the suspended mass will be