Constraint motion and pulleys

Two blocks of masses 2 kg and 1 kg respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in 2 s is ________ m. (Take $g=10$ m/s²)

Three masses $m_1=4$ kg, $m_2=4$ kg and $m_3=6$ kg are suspended from a fixed smooth frictionless pully as shown in the figure below. The value of $T_1/T_2$ is ________.
(take $g=10$ m/s²)

A wedge $Y$ with mass of 10 kg and all frictionless surfaces and the inclined surface making 37° with horizontal. A block $X$ with mass 2 kg is placed at the highest point of the wedge as shown in figure is at rest. At $t=0$ wedge ($Y$) is pulled toward right with constant force ($f$) of 24 N. Taking the block $X$ at rest at $t=0$, the time taken by it to slide down 8.8 m on the slope, while $Y$ is on the move, is ________ s.
(take $\tan(37^\circ)=3/4$ and $g=10$ m/s²)

The velocity at which 6 kg mass (shown in figure) strikes the ground when it is released from a height of 6 m above the ground is ________ m/s. Assume pulley is massless and string is light and inextensible. (Take $g=10$ m/s²)

In the arrangement shown in figure $a_1$, $a_2$, $a_3$ and $a_4$ are the accelerations of masses $m_1$, $m_2$, $m_3$ and $m_4$ respectively. Which of the following relation is true for this arrangement ?

A system to 10 balls each of mass 2 kg are connected via massless and unstretchable string. The system is allowed to slip over the edge of a smooth table as shown in figure. Tension on the string between the 7$^{\rm th}$ and 8$^{\rm th}$ ball is ______ N when 6$^{\rm th}$ ball just leaves the table.

A hanging mass M is connected to a four times bigger mass by using a string-pulley arrangement, as shown in the figure. The bigger mass is placed on a horizontal ice-slab and being pulled by 2 Mg force. In this situation, tension in the string is $\frac x5$ Mg for x = ______. Neglect mass of the string and friction of the block (bigger mass) with ice slab.

(Given g = acceleration due to gravity)

A uniform disc with mass $M=4$ kg and radius $R=10$ cm is mounted on a fixed horizontal axle as shown in figure. A block with mass $m=2$ kg hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is ______ N.

(Take $g=10$ ms$^{-2}$)

A block of mass 40 kg slides over a surface, when a mass of 4 kg is suspended through an inextensible massless string passing over frictionless pulley as shown below.

The coefficient of kinetic friction between the surface and block is 0.02. The acceleration of block is. (Given $g=10$ ms$^{-2}$.)

Three masses M = 100 kg, $m_1$ = 10 kg and $m_2$ = 20 kg are arranged in a system as shown in figure. All the surfaces are frictionless and strings are inextensible and weightless. The pulleys are also weightless and frictionless. A force F is applied on the system so that the mass $m_2$ moves upward with an acceleration of 2 ms$^{-2}$. The value of F is :

(Take $g=10$ ms$^{-2}$)

Two masses $M_1$ and $M_2$ are tied together at the two ends of a light inextensible string that passes over a frictionless pulley. When the mass $M_2$ is twice that of $M_1$, the acceleration of the system is $a_1$. When the mass $M_2$ is thrice that of $M_1$, the acceleration of the system is $a_2$. The ratio $\dfrac{a_1}{a_2}$ will be :

A uniform metal chain of mass m and length 'L' passes over a massless and frictionless pulley. It is released from rest with a part of its length '$l$' is hanging on one side and rest of its length 'L − $l$' is hanging on the other side of the pully. At a certain point of time, when $l=\dfrac Lx$, the acceleration of the chain is $\dfrac g2$. The value of $x$ is ______.

A block of mass $2M$ is attached to a massless spring with spring-constant $k$. This block is connected to two other blocks of masses $M$ and $2M$ using two massless pulleys and strings. The accelerations of the blocks are $a_1$, $a_2$ and $a_3$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $x_0$. Which of the following option(s) is/are correct?

[$g$ is the acceleration due to gravity. Neglect friction]

A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking $g=10\ \mathrm{m/s^2}$, find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest.