Mechanical Properties of Solids

A square aluminum (shear modulus is $25\times10^9$ Nm$^{-2}$) slab of side 60 cm and thickness 15 cm is subjected to a shearing force (on its narrow face) of $18.0\times10^4$ N. The lower edge is riveted to the floor. The displacement of the upper edge is ______ μm.

A steel wire of length 3.2 m ($Y_s=2.0\times10^{11}$ Nm$^{-2}$) and a copper wire of length 4.4 m ($Y_c=1.1\times10^{11}$ Nm$^{-2}$), both of radius 1.4 mm are connected end to end. When stretched by a load, the net elongation is found to be 1.4 mm. The load applied, in Newton, will be : (Given $\pi=\dfrac{22}7$)

In an experiment to determine the Young's modulus, steel wires of five different lengths (1, 2, 3, 4, and 5 m) but of same cross section (2 mm$^2$) were taken and curves between extension and load were obtained. The slope (extension/load) of the curves were plotted with the wire length and the following graph is obtained. If the Young's modulus of given steel wires is $x\times10^{11}$ Nm$^{-2}$, then the value of $x$ is ______.

The force required to stretch a wire of cross-section 1 cm$^2$ to double its length will be :

(Given Yong's modulus of the wire = $2\times10^{11}$ N/m$^2$)

A string of area of cross-section 4 mm$^2$ and length 0.5 m is connected with a rigid body of mass 2 kg. The body is rotated in a vertical circular path of radius 0.5 m. The body acquires a speed of 5 m/s at the bottom of the circular path. Strain produced in the string when the body is at the bottom of the circle is ______ $\times10^{-5}$.

(use young's modulus $10^{11}$ N/m$^2$ and g = 10 m/s$^2$)

If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modulus of the material of the wire will :

A metal wire of length 0.5 m and cross-sectional area $10^{-4}$ m$^2$ has breaking stress $5\times10^8$ N m$^{-2}$. A block of 10 kg is attached at one end of the string and is rotating in a horizontal circle. The maximum linear velocity of block will be _______ m s$^{-1}$.

The speed of a transverse wave passing through a string of length 50 cm and mass 10 g is 60 m s$^{-1}$. The area of cross-section of the wire is 2.0 mm$^2$ and its Young's modulus is $1.2\times10^{11}$ N m$^{-2}$. The extension of the wire over its natural length due to its tension will be $x\times10^{-5}$ m. The value of $x$ is _______.

A block of mass M is attached to a wire. The upper part of wire is made of steel and lower part of wire of copper (as shown in figure). Both parts have the same area of cross section. Neglecting the mass of wires, the ratio of longitudinal strain developed in steel and copper wires is :

(Given, Young's modulus of steel = $2\times10^{11}$ N/m$^2$ and Young's modulus of copper = $1.5\times10^{11}$ N/m$^2$)

If Y, K and η are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

A cubical solid aluminium (bulk modulus $= -V\dfrac{dP}{dV} = 70$ GPa) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be $10^3\ \mathrm{kg\,m^{-3}}$ and $10\ \mathrm{m\,s^{-2}}$, respectively, the change in the edge length of the block in mm is ______.

A block of weight 100 N is suspended by copper and steel wires of same cross sectional area 0.5 cm² and, length $\sqrt3$ m and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30° and 60°, respectively. If elongation in copper wire is $(\Delta l_C)$ and elongation in steel wire is $(\Delta l_S)$, then the ratio $\dfrac{\Delta l_C}{\Delta l_S}$ is ______.

[Young's modulus for copper and steel are $1\times10^{11}\ \mathrm{N/m^2}$ and $2\times10^{11}\ \mathrm{N/m^2}$, respectively.]

A steel wire of diameter 0.5 mm and Young’s modulus $2 \times 10^{11}\ \mathrm{N\,m^{-2}}$ carries a load of mass $M$. The length of the wire with the load is 1.0 m. A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count 1.0 mm, is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by 1.2 kg, the vernier scale division which coincides with a main scale division is __________. Take $g = 10\ \mathrm{m\,s^{-2}}$ and $\pi = 3.2$.

A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of :

An external pressure P is applied on a cube at 0°C so that it is equally compressed from all sides. K is the bulk modulus of the material of the cube and α is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by :

A compressive force, F is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by ΔT. The net change in its length is zero. Let $l$ be the length of the rod, A its area of cross-section, Y its Young's modulus, and α its coefficient of linear expansion. Then, F is equal to :

A steel rail of length 5 m and area of cross section 40 cm$^2$ is prevented from expanding along its length while the temperature rises by 10°C. If coefficient of linear expansion and Young's modulus of steel are $1.2\times10^{-5}$ K$^{-1}$ and $2\times10^{11}$ Nm$^{-2}$ respectively, the force developed in the rail is approximately :

A uniformly tapering conical wire is made from a material of Young's modulus Y and has a normal, unextended length L. The radii, at the upper and lower ends of this conical wire, have values R and 3R, respectively. The upper end of the wire is fixed to a rigid support and a mass M is suspended from its lower end. The equilibrium extended length, of this wire, would equal :

A bottle has an opening of radius a and length b. A cork of length b and radius (a + Δa) where (Δa << a) is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is B and frictional coefficient between the bottle and cork is μ then the force needed to push the cork into the bottle is :

A thin 1 m long rod has a radius of 5 mm. A force of 50 πkN is applied at one end to determine its Young's modulus. Assume that the force is exactly known. If the least count in the measurement of all lengths is 0.01 mm, which of the following statements is false ?