The $40 \mathrm{kg}$ block is moving to the right with a speed of $1.5 \mathrm{m} {\mathrm{s}}^{-1}$ when it is acted upon by the forces $F_1$ and $F_2$. These forces vary in the manner shown in the graph. Find the velocity (in $\mathrm{m} {\mathrm{s}}^{-1}$) of the block at $t=12 \mathrm{s}$. Neglect friction and masses of the pulleys and cords.

 

Two blocks of masses $10 \mathrm{kg}$ and $30 \mathrm{kg}$ are placed along a vertical line. The first block is raised through a height of $7 \mathrm{cm}$. By what distance should the second mass be moved to raise the centre of mass by $1 \mathrm{cm}$.

A rocket of mass, $m=20 \mathrm{kg}$ has $M=180 \mathrm{kg} \mathrm{fuel}.$ The uniform exhaust velocity of the fuel is $v_{\mathit{r}}= 1.6 \mathrm{km} {\mathrm{s}}^{-1}.$
(i) Calculate the minimum rate of consumption of fuel so that the rocket may rise from the ground.
(ii) Also calculate the maximum vertical speed gained by the rocket when the rate of consumption of fuel $\mathrm{\mu }$ is

$$\begin{gathered} \left(\mathrm{a}\right) 2 \mathrm{kg} {\mathrm{s}}^{-1} \\ \left(\mathrm{b}\right) 20 \mathrm{kg} {\mathrm{s}}^{-1} \end{gathered}$$

Use $(g\mathit{ }= 10 \mathrm{m} {\mathrm{s}}^{-2})$ and $(\ln 10 = 2.30)$

A uniform rod $AB$ of length $l$ is released from rest with $AB$ inclined at angle $\mathrm{\theta }$ with horizontal. It collides elastically with smooth horizontal surface after falling through a height $h$. What is the height upto which the centre of mass of the rod rebounds after impact?

Three particles $A, B, C$of mass $m$ each are joined to each other by massless rigid rods to form an equilateral triangle of side $a$. Another particle of mass $m$ hits $B$ with a velocity $v_0$ directed along $BC$ as shown. The colliding particle stops immediately after impact.

(a) Calculate the time required by the triangle $ABC$ to complete half revolution in its subsequent motion.

(b) What is the net displacement of point $B$ during this time interval?

A drinking straw of mass $2 \mathrm{m}$ is placed on a smooth table orthogonally to the edge such that half of it extends beyond the table. A fly of mass $m$ lands on the $A$end of the straw and walks along the straw until it reaches the $B$ end. It does not tip even when another fly gently lands on the top of the first one. Find the largest mass that the second fly can have. (Neglect the friction between straw and table).

Two blocks of masses $m_1=1.0 \mathrm{kg}$ and $m_2=2.0 \mathrm{kg}$ are connected by a massless elastic spring and are at rest on a smooth horizontal surface with the spring at its natural length. A horizontal force of constant magnitude $F= 6.0 \mathrm{N}$ is applied to the block $m_1$ for a certain time $t$ in which $m_1$ suffers a displacement $∆x_1=0.1 \mathrm{m}$and$∆x_2 = 0.05 \mathrm{m}$ . Kinetic energy of the system with respect to centre of mass is $0.1 \mathrm{j}$ force $F$ is then withdrawn.

(a) Calculate $t$

(b) Calculate the speed and the kinetic energy of the centre of mass after the force is withdrawn.

(c) Calculate the energy stored in the system