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,

An ideal spring, with a pointer attached to its end, hangs next to a scale. With $100\mathrm{ }\mathrm{N}$ weight attached and in equilibrium, the pointer indicates $40$ on the scale, as shown. Using a $200\mathrm{ }\mathrm{N}$ weight instead results in $60$ on the scale. Using an unknown weight $X$ instead results in $30$ on the scale. Find the value of $X$.

$\left(a\right)$ Find the value of $\frac{m_1}{m_2}$ in terms of spring constants for this to happen.

$\left(b\right)$ If $k_1=k_2=k_3=500 \mathrm{N} {\mathrm{m}}^{-1}$ and $m_1=2 \mathrm{kg}$ and acceleration of the elevator is $2.5 \mathrm{m} {\mathrm{s}}^{-2}$, find the tension in the middle spring in the final equilibrium with respect to the lift.

The tension in the given spring is

A spring balance and a physical balance are kept in a lift. In these balances, equal masses are placed. Now, if the lift starts moving upwards with constant acceleration, then,

The masses of $10 \mathrm{kg}$ and $20 \mathrm{kg}$, respectively, are connected by a massless spring, as shown in the figure. A force of $200 \mathrm{N}$ acts on the $20 \mathrm{kg}$ mass. At the instant shown, the $10 \mathrm{kg}$ mass has acceleration $12 \mathrm{m} {\mathrm{s}}^{-2}$. What is the acceleration of $20 \mathrm{kg}$ mass?

An ideal spring is compressed and placed horizontally between a vertical fixed wall and a block, free to slide over a smooth horizontal tabletop as shown in the figure. The system is released from rest. The graph which represents the relation between the magnitude of the acceleration $a$ of the block and the distance $x$ travelled by it (as long as the spring is compressed) is,

Two blocks of masses $m_1$ and $m_2$, which are connected with a light string, are placed over a frictionless pulley. This set-up is placed over a weighing machine, as shown. Three combinations of masses $m_1$ and $m_2$ are used. In the first case, $m_1=$ $6 \mathrm{kg}$ and $m_2=2 \mathrm{kg}$, in second case, $m_1=5 \mathrm{kg}$ and $m_2=3 \mathrm{kg}$ and in the third case, $m_1=4 \mathrm{kg}$ and $m_2=4 \mathrm{kg}$. The masses are held stationary initially and then released. If the readings of the weighing machine after the release, in three cases, are $W_1, W_2$ and $W_3$, respectively, then,