Two masses $M$ and $m$ are attached to a vertical axis by weightless threads of combined length $l$. They are set in rotational motion in a horizontal plane about this axis with constant angular velocity $\omega$. If the tensions in the threads are the same during motion,the distance of $M$ from the axis is

$A$ particle of mass $m$ is rotating in a plane in a circular path of radius $r$. Its angular momentum is $L$. The centripetal force acting on the particle is:

One end of a string of length $l$ is connected to a particle of mass $m$ and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed $v$,the net force on the particle (directed towards the centre) is :
$(i) \; T$
$(ii) \; T - \frac{m v^{2}}{l}$
$(iii) \; T + \frac{m v^{2}}{l}$
$(iv) \; 0$
$T$ is the tension in the string. [Choose the correct alternative].

Assume a proton is rotating along a circular path of radius $1 \,m$ under a centrifugal force of $4 \times 10^{-12} \,N$. If the mass of the proton is $1.6 \times 10^{-27} \,kg$, then its angular velocity of rotation is

As shown in the figure,two strings of length $l$ each are attached to a vertical axis $AB$ of length $l$. The strings are $AC$ and $BC$. At point $C$,a point mass $m$ is attached. The mass rotates about the axis with angular velocity $\omega$. The tensions in strings $AC$ and $BC$ are $T_1$ and $T_2$ respectively. Choose the $CORRECT$ alternative:

$A$ car travelling on a smooth road passes through a curved portion of the road in the form of an arc of a circle of radius $10 \, m$. If the mass of the car is $500 \, kg$,the reaction on the car at the lowest point $P$ where its speed is $20 \, m/s$ is ......... $kN$.