DPP

A perfectly straight portion of a uniform rope has mass $M$ and length $L$. At end $A$ of the segment, the tension in the rope is $T_{\mathrm{A}}$ and at the end $B$ it is $T_{\mathrm{B}}\left(T_{\mathrm{B}}>T_{\mathrm{A}}\right)$. Neglect the effect of gravity and no contact force acts on the rope in between points $A$ and $B$. The tension in the rope at a distance $\frac{L}{5}$ from the end $A$

A uniform rope of length $L$ and mass $M$ is placed on a smoothly fixed wedge, as shown. Both ends of the rope are at the same horizontal level. If the rope is initially released from rest, then the magnitude of initial acceleration of the rope will be

In the figure, if block $A$ and wedge $B$ move with the same acceleration then, the magnitude of normal reaction between the block and the wedge will be, (there is no friction between the block and the wedge and the wedge moves on a horizontal surface as shown.)

A block is placed on a smooth wedge of inclination angle $\mathrm{\theta }$ with the horizontal. What minimum acceleration should be given to the wedge in the horizontal direction so that the block starts moving up along the incline?

A rope of negligible mass passes over a pulley of a negligible mass attached to the ceiling, as shown in the figure. One end of the rope is held by a student $A$ of mass $70\mathrm{ }\mathrm{kg}$, who is at rest on the floor. The opposite end of the rope is held by a student $B$ of mass $60\mathrm{ }\mathrm{kg}$, who is suspended at rest above the floor. The minimum acceleration with which the student $B$ should climb up the rope to lift the student $A$ upward, off the floor, is

A balloon is tied to a block. The mass of the block is $2\mathrm{ }\mathrm{kg}$. The tension of the string between the balloon and the block is $30\mathrm{ }\mathrm{N}$. Due to the wind, the string has an angle $\mathrm{\theta }$ relative to the vertical direction and here, $\mathrm{cos\theta }=\frac{4}{5}$ and $\mathrm{sin\theta }=\frac{3}{5}$. Assume that acceleration due to gravity is $g=10\mathrm{ }\mathrm{m} {\mathrm{s}}^{-2}$. Also, assume the block is small so that the force on it due to the wind can be ignored. Then, the $x-$ component and the $y-$ component of the acceleration $a$ of the block,

Two blocks $A$ and $B$ of equal mass $m$ are connected through a massless string and arranged as shown in figure. The wedge is fixed on horizontal surface. Friction is absent everywhere. When the system is released from rest,

In the figure shown, all the surfaces are smooth. All the blocks $A$, $B$ and $C$ are movable, $x$-axis is horizontal and $y$-axis is vertical, as shown. Just after the system is released from the position as shown,