A motor cyclist moves around a circular track with a certain speed and leans at an angle ${\theta }_1$. If he doubles the speed, then he has to lean inward at an angle ${\theta }_2$. Then,

A large mass $M$ hangs stationary at the end of a light string that passes through a smooth fixed tube to a small mass $m$ that moves around in a horizontal circular path. If $l$ is the length of the string from $m$ to the top end of the tube and $\mathrm{\theta }$ is the angle between this part and vertical part of the string, as shown in the figure, then the time taken by $m$to complete one circle is equal to,

A mass $M$ is suspended by a rope from a rigid support at $A$ as shown in figure. Another rope is tied at the end $B$, and it is pulled horizontally with a force $F$. If the rope $AB$ makes an angle $\mathrm{\theta }$ with the vertical in equilibrium, then the tension in the string $AB$ is

One end of the massless rope which passes over a massless and frictionless pulley $P$ is tied to a hook $C$ while the other end is free. The maximum tension that the rope can bear is $360 \mathrm{N}$. With what value of minimum safe acceleration $\left(\mathrm{in} \mathrm{m} {\mathrm{s}}^{-2}\right)$ can a man of $60 \mathrm{kg}$ climb down the rope? (Assume acceleration due to gravity $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.)

Two blocks are connected by a cord passing over a small frictionless pulley and resting on frictionless planes as shown in the figure. The acceleration of the block is (assume $g=10 \mathrm{m} {\mathrm{s}}^{-2}$),

A $50 \mathrm{kg}$ person stands on a $25\mathrm{ }\mathrm{kg}$ platform. He pulls on the rope which is attached to the platform via the frictionless pulleys, as shown in the figure. The platform moves upward at a steady rate if the force with which the person pulls the rope is

A particle of mass $m$ is at rest at the origin at time $t=0$. It is subjected to a force $F\left(t\right)=F_0{e}^{–\mathrm{bt}}$ in the $x$-direction. Its speed $v\left(t\right)$ is depicted by which of the following curves?