Test 6: Statics

Instruction: Select the correct option corresponding to questions given below in the form shared at the bottom

Total Marks: 75

1) A given length 2 s of a uniform chain has to be hung between two points at the same level and the tension has not to exceed the weight of a length $b$ of the chain. Show that the greatest span is given by:

a) $\sqrt{\left(b^{2}-s^{2}\right)} \log {(b-s) /(b+s)}$
b) $\sqrt{\left(b^{2}-s^{2}\right)} \log {(b+s) /(b-s)}$
c) $\sqrt{\left(b^{2}+s^{2}\right)} \log {(b+s) /(b-s)}$ d) $\sqrt{\left(b^{2}-2s^{2}\right)} \log {(b+s) /(b-s)}$

2) The end links of a uniform chain slide along a fixed rough horizontal rod of coefficient of friction $\mu$. Then, the ratio of the maximum span to the length of the chain is given by:

a) $\mu \log \left[\frac{1+\sqrt{\left(1+\mu^{2}\right)}}{\mu}\right]$
b) $\mu \log \left[\frac{1+\sqrt{\left(1+\mu^{2}\right)}}{2\mu}\right]$
c) $2\mu \log \left[\frac{1+\sqrt{\left(1+\mu^{2}\right)}}{\mu}\right]$
d) $\mu \log \left[\frac{1+\sqrt{\left(1+2\mu^{2}\right)}}{\mu}\right]$

3) The end links of a uniform chain of length 21 can slide on two smooth rods in the same vertical plane which are inclined in opposite directions at equal angles $\alpha$ to the vertical. Then, the sag in the middle is given by:

a) $l \tan \frac{\alpha}{2}$
b) $2l \tan \frac{\alpha}{2}$
c) $3l \tan \frac{\alpha}{2}$ d) $l \tan \frac{\alpha}{4}$

4) A heavy string of uniform density and thickness is suspended from two given points in the same horizontal plane. A weight an nth that of the string is attached to its lowest point. Show that if $\theta$ and $\phi$ be the inclination to the vertical of the tangents at the highest and lowest points of the string then $\tan \phi$ is given by:

a) $(1+n) \sec \theta$
b) $(1+n) \cos \theta$
c) $(1+n) \sin \theta$
d) $(1+n) \tan \theta$

5) A regular hexagon $ABCDEF$ consists of six equal rods which are each of weight $W$ and are freely jointed together. The hexagon rests in a vertical plane and $AB$ is in contact with a horizontal table. If $C$ and $F$ be connected by a light string, then its tension is given by:

a) $W / \sqrt{2}$
b) $2W / \sqrt{3}$
c) $W / \sqrt{3}$
d) $4W / \sqrt{3}$

6) $ABCDEF$ is a regular hexagon formed of light rods smoothly jointed at their ends with a diagonal rod $AD$. Four equal forces $P$ act inwards at the middle point of $A B$, $C D$, $D E$, $FA$ at right angles to the respective sides. Find the stress in the diagonal $AD$.

a) $\frac{P}{\sqrt{2}}$
b) $\frac{P}{\sqrt{3}}$
c) $\frac{2P}{\sqrt{3}}$
d) $\frac{3P}{\sqrt{2}}$

7) Two light rods AOC and BOD are smoothly hinged connected by a string of length $2 c \sin \alpha$. The rods rest in a vertical plane, with ends $A$ and $B$ on a smooth horizontal table. A smooth circular disc of radius a and weight $W$ is placed on the rods above $O$ with its plane vertical so that the rods are tangents to the disc. Prove that the tension of the string is:

a) $\frac{1}{2} W\left{(a / c) \operatorname{cosec}^{2} \alpha+\tan \alpha\right}$
b) $\frac{1}{3} W\left{(a / c) \operatorname{cosec}^{2} \alpha+\tan \alpha\right}$
c) $\frac{1}{4} W\left{(a / c) \operatorname{cosec}^{2} \alpha+\tan \alpha\right}$
d) $\frac{1}{5} W\left{(a / c) \operatorname{cosec}^{2} \alpha+\tan \alpha\right}$

8) A heavy uniform rod of length $2 a,$ rests with its ends in contact with two smooth inclined planes of inclination $\alpha$ and $\beta$ to the horizon. If $\theta$ be the inclination of the rod to the horizon, then:

a) $\tan \theta=\frac{1}{5}$ $\alpha-\cot \beta)$
b) $\tan \theta=\frac{1}{4}$ $\alpha-\cot \beta)$
c) $\tan \theta=\frac{1}{2}$ $\alpha-\cot \beta)$
d) $\tan \theta=\frac{1}{3}$ $\alpha-\cot \beta)$

9) A solid hemisphere is supported by a string fixed to a point on its rim and to a point on a smooth vertical wall with which the curved surface of the hemisphere is in contact. If $\theta, \phi$ are the inclinations of the string and the plane base of the hemisphere to the vertical, then:

a) $\tan \phi=\frac{3}{8}+\tan \theta$
b) $\tan \phi=\frac{7}{8}+\tan \theta$
c) $\tan \phi=\frac{5}{8}+\tan \theta$
d) $\tan \phi=\frac{1}{8}+\tan \theta$

10) An endless chain of weight $W$ rests in the form of a circular band round a smooth vertical cone which has its vertex upwards and the vertical angle as $2 \alpha$. The, the tension in the chain due to its weight is given by:

a) $T=(W \cot \alpha) /(\pi)$
b) $T=(W \cot \alpha) /(4 \pi)$
c) $T=(W \cot \alpha) /(3 \pi)$
d) $T=(W \cot \alpha) /(2 \pi)$

11) Two heavy rings of weights $W_1$ and $W_2$ slide on a smooth parabolic wire whose axis is horizontal and plane vertical. The rings are connected by a string passing round a smooth peg at the focus. If the vertical depths below the axis be denoted by $y_1$ and $y_2$ respectively, then in the position of equilibrium:

a) $W_{1} / W_{2}=y_{1} / y_{2}$
b) $W_{1} / W_{2}=2y_{1} / y_{2}$
c) $W_{1} / W_{2}=y_{1} / 2y_{2}$
d) $W_{1} / W_{2}=y_{1} / 3y_{2}$

12) A uniform beam, of length $2a$ rests in equilibriun against a smooth vertical wall and upon a peg at a distance $b$ from the wall. The inclination of the beam to the vertical is given by:

a) $\sin ^{-1}\left(\frac{2b}{a}\right)^{1 / 3}$
b) $\sin ^{-1}\left(\frac{b}{a}\right)^{1 / 2}$
c) $\sin ^{-1}\left(\frac{b}{a}\right)^{1 / 3}$
d) $\sin ^{-1}\left(\frac{b}{a}\right)^{1 / 4}$

13) 6. A smooth hemisperical bowl of diameter a is placed so that its edge touches a smooth vertical wall. A heavy uniform rod is in equilibrium inclined at an angle $60^{\circ}$ to the horizontal with one end resting on the surface of the bowl and the other end against the wall. The, the length of the rod is given by:

a) $\left(a+\frac{a}{\sqrt{(11)}}\right)$
b) $\left(a+\frac{a}{\sqrt{(12)}}\right)$
c) $\left(a+\frac{a}{\sqrt{(13)}}\right)$
d) $\left(a+\frac{a}{\sqrt{(14)}}\right)$

14) A uniform beam of length 2 a rests with its ends on two smooth planes which intersect in a horizontal line. If the inclinations of the plane to the horizontal are $\alpha$ and $\beta(\alpha>\beta),$ show that the inclination $\theta$ of the beam to the horizontal in one of the equilibrium positions is given by:

a) $\tan \theta=\frac{1}{2}(\cot \beta-\cot \alpha)$
b) $\tan \theta=\frac{1}{3}(\cot \beta-\cot \alpha)$
c) $\tan \theta=\frac{1}{4}(\cot \beta-\cot \alpha)$
d) $\tan \theta=\frac{1}{5}(\cot \beta-\cot \alpha)$

15) $A$ weight $W$ is supported on a smooth inclined plane by a given weight $P$, connected with $W$ by means of a string passing round a fixed pulley whose position is given. Then the position of equilibrium of weight $W$ on the plane is given by:

a) $\cos \theta=\frac{W \sin \alpha}{P}$
b) $\cos \theta=\frac{W \sin \alpha}{2P}$
c) $\cos \theta=\frac{W \sin \alpha}{3P}$
d) $\cos \theta=\frac{W \sin \alpha}{4P}$