Torque about a Point

Q.

A constant force $F=3i+2j–4k$ is applied at the point $A(1,–2,2)$. Find the vector moment of $F$ about the point $B(2,–1,3)$.

1. $\left(6i–7j+k\right)$

2. $\left(-2i+6j-10k\right)$

3. $\left(-4i-2j-5k\right)$

4. $Noneoftheabove$

Q. If $\overrightarrow{F}$ is the force acting on a particle having position vector $\overrightarrow{r}$ and $\overrightarrow{\tau }$ be the torque of this force about the origin, then

1. $\overrightarrow{r}.\overrightarrow{\tau }\ne 0$ and $\overrightarrow{F}.\overrightarrow{\tau }=0$
2. $\overrightarrow{r}.\overrightarrow{\tau }>0$ and $\overrightarrow{F}.\overrightarrow{\tau }<0$
3. $\overrightarrow{r}.\overrightarrow{\tau }=0$ and $\overrightarrow{F}.\overrightarrow{\tau }=0$
4. $\overrightarrow{r}.\overrightarrow{\tau }=0$ and $\overrightarrow{F}.\overrightarrow{\tau }\ne 0$

Q. In Fig. the bar is uniform and weighing $500\text{N}$. How large must $W$ be in $\text{N}$ if $T_1$ and $T_2$ are to be equal ?

Q. A solid cube is placed on a horizontal surface. The coefficient of friction between them is $\mu$, where $\mu <1/2$. A variable horizontal force is applied on the cube's upper face, perpendicular to one edge and passing through the mid-point of the edge, as shown in figure. The maximum acceleration with which it can move without toppling is

1. $g(1-2\mu )$
2. $g(1+2\mu )$
3. $\frac{g}{2}(1-2\mu )$
4. $\frac{g}{2}(1+2\mu )$

Q. A cubical block of mass $M$ and edge $a$ slides down a rough inclined plane of inclination $\theta$ with a uniform velocity. The torque of the normal force on the block about its centre has a magnitude

1. zero
2. $Mga$
3. $Mga\sin \theta$
4. $\frac{Mga\sin \theta }{2}$

Q. A tall block of mass $M=50\text{kg}$ and base width $b=1\text{m}$ and height $h=3\text{m}$ is kept on a rough inclined surface with coefficient of friction $\mu =0.8$ as shown in figure. The angle of inclination with the horizontal is ${37}^{\circ }.$ Determine whether the block slides down or topple over. Take $g=10{\text{m/s}}^2$.

1. Block will topple
2. Block will slide
3. Block will slide, then topple
4. Block will neither slide nor topple.

Q. A light rod $AB$ of length $2\text{m}$ is suspended from the ceiling horizontally by means of two vertical wires as shown in Fig. One of the wires is made of steel of cross-section $0.1{\text{cm}}^2$ and the other of brass of cross-section $0.2{\text{cm}}^2$. The Young's modulus of brass is $1.0\times {10}^{11}{\text{Nm}}^{-2}$ and of steel is $2.0\times {10}^{11}{\text{Nm}}^{-2}$. A weight $W$ is hung at point $C$ at a distance $x$ from end $A$. It is found that the stress in the two wires is the same when $x=\frac{n}{3}\text{metre}$. Find the value of $n$.

Q. A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N ? What is the linear acceleration of the rope ? Assume that there is no slipping.

Q. The beam and pans of a balance have negligible mass. An object weighs $W_1$ when placed in one pan and $W_2$ when placed in the other pan. The true weight $W$, of the object is

1. $\sqrt{W_1{W}_2}$
2. $(W_1^{-1}+W_2^{-1})/2$
3. $W_1^2+W_2^2$
4. $\sqrt{(W_1+W_2})$

Q. A homogeneous block having its cross section as a parallelogram of sides $a$ and $b$ as shown, is lying at rest and is in equilibrium on a smooth horizontal surface. Then, find the value of acute angle $\theta$ for which it will be in equilibrium

1. $\theta <{\cos }^{-1}\left(\frac{b}{a}\right)$
2. $\theta \le {\cos }^{-1}\left(\frac{b}{a}\right)$
3. $\theta \ge {\cos }^{-1}\left(\frac{b}{a}\right)$
4. $\theta >{\cos }^{-1}\left(\frac{b}{a}\right)$

Q. A regular hexagonal uniform block of mass $m=4\sqrt{3}\text{kg}$ rests on a rough horizontal surface with coefficient of friction $\mu$ as shown in figure. A constant horizontal force is applied on the block as shown. If the friction is sufficient to prevent slipping before toppling, then the minimum force (in $\text{N}$) required to topple the block about its corner $\text{A}$ is
(Take $g=10{\text{m/s}}^2$)

Q. A regular hexagonal uniform block of mass $m=4\sqrt{3}\text{kg}$ rests on a rough horizontal surface with coefficient of friction $\mu$ as shown in figure. A constant horizontal force is applied on the block as shown. If the friction is sufficient to prevent slipping before toppling, then the minimum force (in $\text{N}$) required to topple the block about its corner $\text{A}$ is
(Take $g=10{\text{m/s}}^2$)

1. $5\text{N}$
2. $25\text{N}$
3. $10\text{N}$
4. $20\text{N}$

Q. A solid cube of side ${}^{\prime }{a}^{\prime }$ is shown in the figure. Find maximum value of $100\frac{b}{a}$ for which the block does not topple before sliding.

Q. A box of dimensions $l\times b$ is kept on a truck moving with an acceleration $a$. If the box does not slide, maximum acceleration of the truck for it to remain in equilibrium (w.r.t. the truck) is

1. $\frac{gl}{b}$
2. $\frac{gb}{l}$
3. $g$
4. None of these

Q. A uniform cube of side $a$ and mass $m$ rests on a horizontal table. A horizontal force ${}^{\prime }{F}^{\prime }$ is applied normal to one of the faces at a point that is directly above the centre of the face, at a height $\frac{3a}{4}$ above the base. The minimum value of ${}^{\prime }{F}^{\prime }$ for which the cube begins to tilt about the edge is (assume that the cube does not slide)

1. $\frac{2}{3}mg$
2. $\frac{4}{3}mg$
3. $\frac{5}{4}mg$
4. $\frac{1}{2}mg$