Mechanical Properties of Solids

The diameter of a wire measured by a screw gauge of least count 0.001 cm is 0.08 cm. The length measured by a scale of least count 0.1 cm is 150 cm. When a weight of 100 N is applied to the wire, the extension in length is 0.5 cm, measured by a micrometer of least count 0.001 cm. The error in the measured Young's modulus is $\alpha\times10^9$ N/m². The value of $\alpha$ is ________.
(Ignore the contribution of the load to Young's modulus error calculation)

A uniform wire of length $l$ of weight $w$ is suspended from the roof with a weight of $W$ at the other end. The stress in the wire at $\dfrac l3$ distance from the top is $\left(\dfrac WA+\dfrac2\gamma\dfrac wA\right)$, where, $A$ is the cross sectional area of the wire. The value of $\gamma$ is ________.

The increase in the pressure required to decrease the volume ($\Delta V$) of water is $6.3\times10^7$ N/m². The percentage decrease in the volume is ________.
(Bulk modulus of water $=2.1\times10^9$ N/m².)

A string $A$ of length 0.314 m and Young's modulus $2\times10^{10}$ N/m² is connected to another string $B$ of length and Young's modulus both twice of those of $A$. This series combination of strings is then suspended from a rigid support and its free end is fixed to a load of mass 0.8 kg. The net change in length of the combination is ________ mm.
(radius of both the strings is 0.2 mm and acceleration due to gravity = 10 m/s²)
(Mass of both strings is to be neglected as compared to the mass of load)

A metal string $A$ is suspended from a rigid support and its free end is attached to a block of mass $M$. Second block having mass $2M$ is suspended at the bottom of the first block using a string $B$. The area of cross sections of strings $A$ and $B$ are same. The ratio of lengths of strings of $A$ to B is 2 and the ratio of their Young's moduli ($Y_A/Y_B$) is 0.5. The ratio of elongations in $A$ to B is ________.

The Young's modulus of steel wire of radius $r$ and length $L$ is $Y$.

If the radius $r$ and length $L$ of the wire are doubled then the value of $Y$

Two wires as shown in the figure below, made of steel and have breaking stress of $12\times10^8$ N/m². Area of cross-section of upper wire is 0.008 cm² and of lower wire is 0.004 cm². The maximum mass that can be added to pan without breaking any wire is ________ kg.
(take $g=10$ m/s²)

A cube has side length 5 cm and modulus of rigidity $10^5$ N/m². The displacement produced by a force of 10 N in the upper face of cube is ________ mm.

A copper wire of length 3 m is stretched by 3 mm by applying an external force. The volume of the wire is $600\times10^{-6}$ m³. The elastic potential energy stored in the wire in stretched condition would be ________ J.

(Given Young modulus of copper $=1.1\times10^{11}$ N/m²)

A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the wire can withstand is $4\times10^8$ N/m² and its radius is 4 mm, then maximum acceleration the lift can take is ________ m/s².

(take $g=10$ m/s² and $\pi=3.14$)

The two wires $A$ and $B$ of equal cross-section but of different materials are joined together. The ratio of Young's modulus of wire $A$ and wire $B$ is 20/11. When the joined wire is kept under certain tension the elongations in the wires $A$ and $B$ are equal. If the length of wire $A$ is 2.2 m, then the length of wire $B$ is ________ m.

A certain gas is isothermally compressed to $\left(\dfrac13\right)^{\text{rd}}$ of its initial volume ($V_o=3$ litre) by applying required pressure. If the bulk modulus of the gas is $3\times10^5$ N/m², the magnitude of work done on the gas is ________ J.

Figure represents the extension ($\Delta l$) of a wire of length 1 meter, suspended from the ceiling of the room at one end with a load $W$ connected to the other end. If the cross-sectional area of the wire is $10^{-5}$ m² then the Young's modulus of the wire is ________ N/m².

The bulk modulus of a liquid is $3\times10^{10}\ \mathrm{Nm^{-2}}$. The pressure required to reduce the volume of liquid by 2% is :

The elastic behaviour of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of $5\times10^{-4}$ is ______ kJ/m$^3$. Assume that material is elastic upto the linear strain of $5\times10^{-4}$.

The elongation of a wire on the surface of the earth is $10^{-4}$ m. The same wire of same dimensions is elongated by $6\times10^{-5}$ m on another planet. The acceleration due to gravity on the planet will be ______ ms$^{-2}$. (Take acceleration due to gravity on the surface of earth = 10 ms$^{-2}$)

A wire of length L is hanging from a fixed support. The length changes to $L_1$ and $L_2$ when masses 1 kg and 2 kg are suspended respectively from its free end. Then the value of L is equal to :

A wire of length L and radius r is clamped rigidly at one end. When the other end of the wire is pulled by a force F, its length increases by 5 cm. Another wire of the same material of length 4L and radius 4r is pulled by a force 4F under same conditions. The increase in length of this wire is ______ cm.

The area of cross section of the rope used to lift a load by a crane is $2.5\times10^{-4}$ m$^2$. The maximum lifting capacity of the crane is 10 metric tons. To increase the lifting capacity of the crane to 25 metric tons, the required area of cross section of the rope should be :

(take g = 10 ms$^{-2}$)

A uniform heavy rod of mass 20 kg, cross sectional area 0.4 m$^2$ and length 20 m is hanging from a fixed support. Neglecting the lateral contraction, the elongation in the rod due to its own weight is $x\times10^{-9}$ m. The value of $x$ is ______.

(Given, young modulus Y = $2\times10^{11}$ Nm$^{-2}$ and g = 10 ms$^{-2}$)