Torque about a Point

Q. Three particles, each of mass $m$ and carrying a charge $q$ are suspended from a common point by insulating massless strings, each of length $L$. If the particles are in equilibrium and are located at the corners of an equilateral triangle of side $a$, then calculate the charge $q$ on each particle.
[Assume $L>>a$].

1. ${\left[\frac{2\pi {ϵ}_0{a}^{3}mg}{3L}\right]}^{\frac{1}{2}}$
2. None
3. ${\left[\frac{4\pi {ϵ}_0{a}^{3}mg}{3L}\right]}^{\frac{1}{3}}$
4. ${\left[\frac{4\pi {ϵ}_0{a}^{3}mg}{3L}\right]}^{\frac{1}{2}}$

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 uniform rod of mass $m$ and length $l$ can rotate in a vertical plane about a smooth horizontal axis hinged at point $H$. Find the force exerted by the hinge just after the rod is released as shown in the figure.

1. $mg\sqrt{7}$
2. $\frac{mg\sqrt{10}}{5}$
3. $\frac{mg}{5}$
4. $\frac{mg}{\sqrt{5}}$

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 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. $\frac{1}{2}mg$

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. 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. Moment of the force, $\overrightarrow{F}=(4\hat{i}+5\hat{j}-6\hat{k})\text{N}$ acting at point $P(2,0,-3)\text{m}$ about the point $Q(2,-2,-2)\text{m}$ is given by :

1. $(-7\hat{i}-8\hat{j}-4\hat{k})\text{N.m}$
2. $(-4\hat{i}-\hat{j}-8\hat{k})\text{N.m}$
3. $(-8\hat{i}-4\hat{j}-7\hat{k})\text{N.m}$
4. $(-7\hat{i}-4\hat{j}-8\hat{k})\text{N.m}$

Q. What is the torque produced by the hinge forces on the body about the point O?

1. 4 N-m
2. 3 N-m
3. 5 N-m
4. 10 N-m

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

1. $\mu >\frac{1}{4}$
2. $\mu <\frac{1}{2}$
3. $\mu >\frac{1}{2}$
4. $\mu <1$

Q. The door of an almirah is $6\text{ft}$ high, $1.5\text{ft}$ wide and weighs $8\text{kg}$. The door is supported by two hinges situated at a distance of $1\text{ft}$ from the top and bottom ends. Assuming force exerted on the hinges are equal, the magnitude of the force is
[Take $g=10{\text{m/s}}^2$]

1. $15\text{N}$
2. $10\text{N}$
3. $28\text{N}$
4. $43\text{N}$

Q. A solid hemisphere of radius $R$ is placed on an inclined plane of inclination $\theta .$ What will be the maximum value of $\theta$ for which the hemisphere will not topple? (Assume that the solid hemisphere will not slide)

1. $\theta ={\tan }^{-1}\left(\frac{2R}{\pi }\right)$
2. $\theta ={\tan }^{-1}\left(\frac{3\pi }{4}\right)$
3. $\theta ={\tan }^{-1}\left(2\right)$
4. $\theta ={\tan }^{-1}\left(\frac{8}{3}\right)$

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. Minimum value of $F$ for which the cube begins to tip about an edge is equal to $\frac{2mg}{M}$. Then, $M$ is -

1. $2$
2. $4$
3. $5$
4. $3$

Q. If $\overrightarrow{F}$ be a force acting on a particle having the position vector $\overrightarrow{r}$ and $\overrightarrow{\tau }$ be the torque of this force about the origin, then, $\overrightarrow{\tau }.\overrightarrow{F}=0$ and $\overrightarrow{r}.\overrightarrow{\tau }=0$.

1. False
2. True

Q. The door of an almirah is $6\text{ft}$ high, $1.5\text{ft}$ wide and weighs $8\text{kg}$. The door is supported by two hinges situated at a distance of $1\text{ft}$ from the top and bottom ends. Assuming force exerted on the hinges are equal, the magnitude of the force is
[Take $g=10{\text{m/s}}^2$]

1. $15\text{N}$
2. $10\text{N}$
3. $28\text{N}$
4. $43\text{N}$

Q. Minimum value of $F$ for which the cube begins to tip about an edge is equal to $\frac{2mg}{M}$. Then, $M$ is -

1. $2$
2. $4$
3. $5$
4. $3$

Q. Moment of the force, $\overrightarrow{F}=(4\hat{i}+5\hat{j}-6\hat{k})\text{N}$ acting at point $P(2,0,-3)\text{m}$ about the point $Q(2,-2,-2)\text{m}$ is given by :

1. $(-7\hat{i}-8\hat{j}-4\hat{k})\text{N.m}$
2. $(-4\hat{i}-\hat{j}-8\hat{k})\text{N.m}$
3. $(-8\hat{i}-4\hat{j}-7\hat{k})\text{N.m}$
4. $(-7\hat{i}-4\hat{j}-8\hat{k})\text{N.m}$

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. $\frac{Mga\sin \theta }{2}$