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A steel rod of length l, area of coss section A, Young's modulus Y and linear coefficient of expansion $\alpha$ is heated through $t°C$. The work that can be performed by the rod when heated is:
(1) $\frac{1}{2}(YA \alpha t)\times (l \alpha t)$
(2) $(YA \alpha t)\times (l \alpha t)$
(3) 2$(YA \alpha t)\times (l \alpha t)$
(4) $\frac{1}{2}(YA \alpha t)\times (\frac{1}{2}l \alpha t)$

A wire of cross-sectional area A is stretched horizontally between two clamps located at a distance l metres from each other. A weight W kg is suspended from the mid point of the wire. If the vertical distance through which the mid point of the wire moves down be x<l, then the strain produced in the wire is:
(1) $x^2/\left(4l^2\right)$
(2) $x^2/(2 l^2)$
(3) $x^2/l^2$
(4) 2 $x^2/l^2$

A constant force$F_0$ is applied on a uniform elastic string placed over a smooth horizontal surface as shown in figure. Young's modulus of string is Y and area of cross-section is A. The strain produced in the string in the direction of force is:

(1) $\frac{F_{0}Y}{A}$
(2) $\frac{F_0}{AY}$
(3) $\frac{F_0}{2AY}$
(4) $\frac{F_{0}Y}{2A}$

A thin uniform metallic rod of length L and area of cross-section A rotates with an angular velocity $\omega$ in a horizontal plane about a vertical axis passing through one of its ends. If $\rho$ is the density of the rod, the maximum tension in the rod is:
(1) $\frac{1}{2}\rho A{\omega }^2{L}^2$
(2) $\frac{1}{2}\rho A{\omega }^{2}L$
(3) $\frac{1}{3}\rho A{\omega }^2{L}^2$
(4) $\frac{1}{3}\rho A{\omega }^{2}L$

Two equal and opposite forces F and -F act on a rod of uniform cross section area A as shown in figure. The longitudinal stress on the section AB is:

(1) $\frac{F}{A}$
(2) $\frac{2F}{A}$
(3) $\frac{F}{A}\sin \theta$
(4) $\frac{F}{A}{\sin }^2\theta$

Two cylinders A and B of radii r and 2r are soldered co-axially. The free end of A is clamped and free end of B is twisted by an angle $\varphi$. The twist at the junction, taking the material of two cylinders to be same and length equal, is:
(1) $\frac{15}{16}\varphi$
(2) $\frac{16}{17}\varphi$
(3) $\frac{16}{15}\varphi$
(4) $\frac{17}{16}\varphi$

Two wires of equal length and  cross-section area suspended as shown in figure. Their Young's modulus are $Y_1 and Y_2$ respectively. The equivalent Young's modulus will be:

(1) $Y_1+Y_2$
(2) $\frac{Y_1+Y_2}{2}$
(3) $\frac{Y_1{Y}_2}{Y_1+Y_2}$
(4) $\sqrt{Y_1{Y}_2}$

If the ratio of lengths, radii and Young's modulii of steel and brass wires in the figure are a, b, c respectively. Then the corresponding ratio of increase in their lengths would be:

(1) $\frac{2ac}{b^2}$
(2) $\frac{3a}{2b^{2}c}$
(3) $\frac{3c}{2a{b}^2}$
(4) $\frac{2a^{2}c}{b}$

A steel ring of radius r and cross sectional area A is fitted on to a wooden disc of radius R(R>r). If Young's modulus be Y, then the force with which the steel ring is expanded, is:
(1) $AY\frac{R}{r}$
(2) $AY\left[\frac{R-r}{r}\right]$
(3) $\frac{Y}{A}\left[\frac{R-r}{r}\right]$
(4) $\frac{Yr}{AR}$

Two particles of equal mass go round a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is:
(1) $V=\sqrt{\frac{Gm}{2R}}$
(2) $V=\frac{1}{2R}\sqrt{\frac{1}{Gm}}$
(3) $V=\frac{1}{2}\sqrt{\frac{Gm}{R}}$
(4) $V=\sqrt{\frac{4Gm}{R}}$

One end of a uniform rod of mass ${\mathrm{m}}_1$, uniform area of cross-section A is suspended from the roof and a mass ${\mathrm{m}}_2$ is suspended from the other end. What is the stress at the mid point of the rod?
1. $({\mathrm{m}}_1+{\mathrm{m}}_2) \mathrm{g}/\mathrm{A}$
2. $({\mathrm{m}}_1-{\mathrm{m}}_2) \mathrm{g}/\mathrm{A}$
3. $\left[\frac{({\mathrm{m}}_1/2) +{\mathrm{m}}_2}{\mathrm{A}}\right]$
4. $\left[\frac{{\mathrm{m}}_1+({\mathrm{m}}_2/2)}{\mathrm{A}}\right]g$

Two spheres of radius r and 2r are touching each other. The force of attraction between them is proportional to
1. ${\mathrm{r}}^6$
2. ${\mathrm{r}}^4$
3. ${\mathrm{r}}^2$
4. ${\mathrm{r}}^{-2}$