Equilibrium of Rigid Bodies

A rectangular block has a square base measuring area $a^2$ and its height is $h$ . It moves on a horizontal surface in a direction perpendicular to one of the edges. The coefficient of friction is $\mu$. It will topple if

An ice cube of mass $M$ and with sides of length a is sliding without friction with speed $v_0$ when it hits a ridge $O$ at the edge of counter (see figure). This collision causes the cube to tilt as shown. The minimum value of ${\mathrm{v}}_0$ needed for the cube to fall off the table is given by $\sqrt{\mathrm{kag}}.$ Find value of $k.$ (Approximate the answer to nearest integer.)

Consider a uniform cubical box of side $a$ on a rough floor that is to be moved by applying minimum possible force $F$ at a point $b$ above its centre of mass (see figure). If the coefficient of friction is $\mu =0.4,$ the maximum possible value of $100\times \frac{\mathrm{b}}{\mathrm{a}}$ for box not to topple before moving is ________

A body is acted upon by two unequal forces in opposite directions, but not in the same line. The effect is that 

A$\text{'}T\text{'}$ shaped object with dimensions shown In the figure, is lying on a smooth floor. A force $\text{'}F\text{'}$ is applied at the point $P$ parallel to $AB\mathit{,}$ such that the object has only the translational motion without rotation. Find the location of $P$ with respect to $C\mathit{ }\mathit{:}$

A rod of mass $m$ and length $l$ is suspended from a vertical wall and kept horizontal by a massless vertical thread as shown. The tension in the thread is,

A solid cylinder of mass $m$ is wrapped with a string and placed on a rough inclined plane as shown in the fig. Then the frictional force acting between cylinder and plane is

A uniform cube of side a & mass m rests on a rough horizontal table. A horizontal force $F$ is applied normal to one of the faces at a point that is directly above the centre of the face, at a height $3 a/4$ above the base. The minimum value of $F$ for which the cube begins to tip about an edge is -----.(assume that cube does not slide).

A non-uniform bar of weight $W$ is suspended at rest by two strings of negligible weight as shown in Fig. The angles made by the strings with the vertical are $36.9°$ and $53.1°$ respectively. The bar is $2 \mathrm{m}$ long. Calculate the distance $d$ of the centre of gravity of the bar from its left end.

A spool is pulled at an angle $\theta$ with the horizontal on a rough horizontal surface as shown in the figure. If the spool remains at rest, the angle $\theta$ is equal to

A uniform rod of mass $m$ is leaning at rest against a smooth wall as shown. Contact force between the rod and the ground is $\left[\cot \theta =2\sqrt{3}\right]$

A rod of mass $M$ and length $L$ is hinged about one end and a particle of mass $M$ is attached to its other end. If it is kept horizontal with the help of a spring of spring constant $K$ as shown in the figure. Find the extension in spring:-

A weightless rod is acted upon by upward parallel forces of $2 \mathrm{N}$ and $4 \mathrm{N}$ at ends $A$ and $B$ respectively.
The total length of the $rod AB=3 \mathrm{m}.$ To keep the rod in equilibrium a force of $6 \mathrm{N}$ should act in the following manner:

A uniform rod of $20 \mathrm{kg}$ is hanging in a horizontal position with the help of two threads. It also supports a $40 \mathrm{kg}$ mass as shown in the figure. The tensions developed in each thread will be

An engineer is designing a conveyor system for loading lay bales into a wagon. Each bale is 0.25 m wide, 0.50 m high, and 0.80 m long (the dimension perpendicular to the plane of the figure), with mass 30.0 kg. The center of gravity of each bale is at its geometrical center. The coefficient of static friction between a bale and the conveyor belt is 0.60, and the belt moves with constant speed. The angle $\text{β}$ of the conveyor is slowly increased. At some critical angle a bale will tip (if it doesn't slip first), and at some different critical angle it will slip (if it doesn't tip first).

Find the first critical angle (In the same conditions) at which it tips..

A thin uniform rod of mass m and length $L$ is moving translationally with acceleration $\text{'}a\text{'}$ due to two antiparallel forces, as shown. One of the forces is $F$. Find the maximum value of acceleration for which rod may move translationally.

A rod of mass $\mathrm{M}$ hinged at $\mathrm{O}$ is kept in equilibrium with a spring of stiffness $\mathrm{k}$ as shown in figure. The potential energy stored in the spring is

Two light vertical springs with spring constants $k_1$ and $k_2$ are separated by a distance $l$. Their upper ends are fixed to the ceiling and their lower ends to the ends $A$ and $B$ of a light horizontal rod $\mathrm{AB}$. A vertical  downward force $F$ is applied at point $C$ on the rod. $AB$ will remain horizontal in equilibrium if the distance $AC$ is,

A uniform cube of mass $m$ and side $a$ is resting in equilibrium on a rough $45°$ inclined surface. The distance of the point of application of normal reaction measured from the lower edge of the cube is

A force $F$ is applied on the top of a cube as shown in the figure. The coefficient of friction between the cube and ground is $\mu$. If  $F$ is gradually increased, then the cube will topple before sliding. Then the range of $\mu$ is