Torque about a Point

Q. A wire of area of cross-section ${10}^{-6}{\text{m}}^2$ is increased in length by $0.1\%$. The tension produced is $1000\text{N}$. The Young’s modulus of wire is

1. ${10}^{11}\text{N}/{\text{m}}^2$
2. ${10}^9\text{N}/{\text{m}}^2$
3. ${10}^{10}\text{N}/{\text{m}}^2$
4. ${10}^{12}\text{N}/{\text{m}}^2$

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. $\overrightarrow{r}.\overrightarrow{\tau }=0$ and $\overrightarrow{F}.\overrightarrow{\tau }\ne 0$
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. $\overrightarrow{r}.\overrightarrow{\tau }\ne 0$ and $\overrightarrow{F}.\overrightarrow{\tau }=0$

Q.

Three forces $\mathrm{P},\mathrm{Q},\mathrm{R}$ act along the sides $\mathrm{BC},\mathrm{CA},\mathrm{AB}$ of triangle $\mathrm{ABC}$, taken in order. If their resultant passes through the incentre of$∆\mathrm{ABC}$, then

1. $\mathrm{P}+\mathrm{Q}+\mathrm{R}=0$

2. $\left(\raisebox{1ex}{$\mathrm{P}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{a}$}\right.\right)+\left(\raisebox{1ex}{$\mathrm{Q}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{b}$}\right.\right)+\left(\raisebox{1ex}{$\mathrm{R}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{c}$}\right.\right)$

3. $\mathrm{aP}+\mathrm{bQ}+\mathrm{cR}=0$

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

Q.

A wheel having a moment of inertia $2\mathrm{kg}-{\mathrm{m}}^2$ about its vertical axis rotates at the rate of $60\mathrm{rpm}$ about this axis. The torque which can stop the wheels rotation in one minute would be

1. $\frac{2\mathrm{\pi }}{13}\mathrm{N}-\mathrm{m}$

2. $\frac{\mathrm{\pi }}{14}\mathrm{N}-\mathrm{m}$

3. $\frac{\mathrm{\pi }}{15}\mathrm{N}-\mathrm{m}$

4. $\frac{\mathrm{\pi }}{20}\mathrm{N}-\mathrm{m}$

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. $3$
4. $5$

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 solid cylindrical roller of radius $R$ and mass$m$ is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is:

1. $\frac{2F}{3mR}$

2. $\frac{3F}{2mR}$

3. $\frac{F}{3mR}$

4. $\frac{F}{2mR}$

Q. Three identical small balls, each of mass $0.1\text{g}$ are suspended at one point on the silk thread having a length of $20\text{cm}$. What charges should be imparted to the balls for each thread to form an angle of $\alpha ={30}^{\circ }$ with the vertical ?

1. $1.1\times {10}^{-8}\text{C}$
2. $2.2\times {10}^{-8}\text{C}$
3. $3.3\times {10}^{-8}\text{C}$
4. $4.4\times {10}^{-8}\text{C}$

Q.

Suppose the rod in the previous problem has a mass of 1 kg distributed uniformly over its length.

(a) Find the initial angular acceleration of the rod.

(b) Find the tension in the supports to the blocks of mass 2 kg and 5 kg.

Q. An equilateral uniform prism of mass $m$ rests on a rough horizontal surface with coefficient of friction $\mu$ and a horizontal force $F$ is applied. If the coefficient of friction is sufficiently high so that the prism does not slide before toppling, then the minimum force required to topple the prism is:

1. $\frac{mg}{\sqrt{3}}$
2. $\frac{mg}{4}$
3. $\frac{\mu mg}{\sqrt{4}}$
4. $\frac{\mu mg}{\sqrt{3}}$

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

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 100 N force acts horizontally on a block of mass 10 kg placed on a horizontal rought table of coefficient of friction $\mu =\mathrm{0.5.}$ If g at the place is $10m{s}^{-2},$ the acceleration of the block is

1. Zero
2. $10m/s^2$
3. $5m/s^2$
4. $5.2m/s^2$

Q. A cylinder and a wedge of same mass (m) with a vertical face, touching each other, move along two smooth inclined planes forming an angle and respectively with horizontal as shown in the figure. Determine the force of normal N (In newton) exerted by the wedge on the cylinder, neglecting the friction between them. Consider $m=\frac{1}{\sqrt{3}}kg;\alpha ={60}^{\circ };\beta ={30}^{\circ }andg=10m/s^2.$ ___

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

Q. A cylindrical vessel filled with water is released on an inclined surface of inclination angle $\theta$ as shown in figure. The coefficient of friction between surface with vessel is $\mu (<tan\theta )$. Then the constant angle made by the surface of water with the incline will be

1. $ta{n}^{-1}\mu$
2. $\theta -ta{n}^{-1}\mu$
3. $\theta +ta{n}^{-1}\mu$
4. $co{t}^{-1}\mu$

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 equlibrium

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

Q. A cube of side $a$ is placed on an inclined plane of inclination $\theta .$ What is the maximum value of $\theta$ for which the cube will not topple?

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${15}^{\circ }$

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 bar having a cross-sectional area of 700 $m{m}^2$ is subjected to axial loads at the positons indicated. The value of stress in the segment QR is

1. 40 Mpa
2. 50 Mpa
3. 70 Mpa
4. 120 Mpa

Q. A light rod of length $2\text{m}$ is suspended from the ceiling horizontally by means of two vertical wires of equal length tied to it ends. One of the wire is made of steel and it's cross section is ${10}^{-3}{\text{m}}^2$ and the other is of brass of cross section $2\times {10}^{-3}{\text{m}}^2$. The position along the rod at (from the end where brass wire is tied) which a weight may be hung to produce equal stress on both wires $(Y_{steel}=2\times {10}^{11}{\text{Nm}}^{-2};Y_{brass}={10}^{11}{\text{Nm}}^{-2})$

1. $1\text{m}$
2. $\frac{4}{3}\text{m}$
3. $\frac{2}{3}\text{m}$
4. $\frac{1}{3}\text{m}$

Q. A uniform rod is hinged at one end to a wall and hangs through a rope attached to the other end as shown in the figure. Mass of rod is 80 kg and a carpenter of mass 60 kg is sitting at $\frac{3l}{4}$ from the hinged end. Tension in supporting rope is

1. 1200 N
2. 850 N
3. 2400 N
4. 1700 N

Q. A wire of length $1\text{m}$ and area of cross-section $4\times {10}^{-8}{\text{m}}^2$ increase in length by $0.2\text{cm}$ when a force of $16\text{N}$ is applied. The value of $Y$ for the material of the wire will be

1. $2\times {10}^6{\text{N/m}}^2$
2. $2\times {10}^5{\text{N/m}}^2$
3. $2\times {10}^{12}{\text{N/m}}^2$
4. $2\times {10}^{11}{\text{N/m}}^2$

Q. Which of the following plan views of a gusseted base plate will result in minimum base plate thickness?

1. 
2. 
3. 
4. 

Q. A cube is placed on an inclined plane of inclination $\theta$ as shown in figure. Coefficient of friction between the cube and the plane is $\mu$. As the angle $\theta$ is gradually increased, the cube slides before toppling if

1. $\mu >1$
2. $\mu <1$
3. None of these
4. $\mu >\frac{1}{2}$

Q. A sonometer wire of length $1.5\text{m}$ is made of steel. The tension in it produces an elastic strain of $1\%$. What is the fundamental frequency of steel, if density and elasticity of steel are $7.7\times {10}^3{\text{kg/m}}^3$ and $2.2\times {10}^{11}{\text{N/m}}^2,$ respectively?

1. $188.5\text{Hz}$
2. $178.2\text{Hz}$
3. $250.5\text{Hz}$
4. $770\text{Hz}$

Q. A mass of $10\text{kg}$ is suspended vertically by a rope from the roof. When a horizontal force is applied on the rope at some point, the rope deviated at an angle ${45}^{\circ }$ at the roof point. If the suspended mass is at equilibrium, the magnitude of the force applied is ($g=10{\text{ms}}^{-2}$)

1. $\text{200 N}$
2. $\text{140 N}$
3. $\text{70 N}$
4. $\text{100 N}$

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.