Wire Combinations & Special Cases

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 2 M 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 ______.

A string $A$ of length 0.314 m and Young's modulus $2 \times 10^{10}\text{ N/m}^2$ 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 $= 10\text{ m/s}^2$) (Mass of both strings is to be neglected as compared to the mass of load)

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 $10^{-5}\text{ m}^2$ then the Young's modulus of the wire is $\underline{\qquad}\ \text{N/m}^2$.

The length of a light string is 1.4 m when the tension on it is 5 N. If the tension increases to 7 N, the length of the string is 1.56 m. The original length of the string is__________m.

Two persons pull a wire towards themselves. Each person exerts a force of $200\text{ N}$ on the wire. Young's modulus of the material of wire is $1 \times 10^{11}\text{ N m}^{-2}$. Original length of the wire is $2\text{ m}$ and the area of cross section is $2\text{ cm}^2$. The wire will extend in length by _________ $\mu\text{m}$.

An elastic spring under tension of $3\text{ N}$ has a length $a$. Its length is $b$ under tension $2\text{ N}$. For its length $(3a - 2b)$, the value of tension will be _______ N.

A spring has length L and force constant K. It is cut into two springs of length $L_1$ and $L_2$ such that $L_1 = NL_2$ ( N is an integer). The force constant of spring of length $L_1$ is

A wire of cross sectional area A, modulus of elasticity $2 \times 10^{11}\text{ Nm}^{-2}$ and length $2\text{ m}$ is stretched between two vertical rigid supports. When a mass of $2\text{ kg}$ is suspended at the middle it sags lower from its original position making angle $\theta = \frac{1}{100}$ radian on the points of support. The value of A is ________ $\times 10^{-4}\text{ m}^2$ (consider $x << \text{L}$ ).

(given: $\text{g} = 10\text{ m/s}^2$)

Young's modulus is determined by the equation given by $\text{Y} = 49000\frac{\text{m}}{\text{l}}\,\frac{\text{dyne}}{\text{cm}^2}$ where $M$ is the mass and $l$ is the extension of wire used in the experiment. Now error in Young modules $(Y)$ is estimated by taking data from $M - l$ plot in graph paper. The smallest scale divisions are $5\text{ g}$ and $0.02\text{ cm}$ along load axis and extension axis respectively. If the value of $M$ and $l$ are $500\text{ g}$ and $2\text{ cm}$ respectively then percentage error of $Y$ is:

Two wires each of radius 0.2 cm and negligible mass, one made of steel and the other made of brass are loaded as shown in the figure. The elongation of the steel wire is __________ $\times$ 10 ${}^{-6}$ m. [Young's modulus for steel = 2 $\times$ 10${}^{11}$ Nm${}^{-2}$ and g = 10 ms${}^{-2}$ ]

The length of a wire becomes $l_1$ and $l_2$ when $100\text{ N}$ and $120\text{ N}$ tensions are applied respectively. If $10\,l_2 = 11\,l_1$, the natural length of wire will be $\frac{1}{x}\,l_1$. Here the value of $x$ is _____________.

As shown in the figure, in an experiment to determine Young's modulus of a wire, the extension-load curve is plotted. The curve is a straight line passing through the origin and makes an angle of 45 $^\circ$ with the load axis. The length of wire is 62.8 cm and its diameter is 4 mm. The Young's modulus is found to be $x \times 10^{4}$ Nm$^{-2}$. The value of $x$ is ___________.

A steel wire of length $3.2\text{ m} \left(Y_s = 2.0 \times 10^{11}\text{ N m}^{-2}\right)$ and a copper wire of length $4.4\text{ m} \left(Y_c = 1.1 \times 10^{11}\text{ N m}^{-2}\right)$, both of radius $1.4\text{ mm}$ are connected end to end. When stretched by a load, the net elongation is found to be $1.4\text{ mm}$. The load applied, in Newton, will be: $($ Given $\pi = \frac{22}{7}$)

A uniform heavy rod of mass $20\text{ kg}$, cross sectional area $0.4\text{ m}^2$ and length $20\text{ m}$ is hanging from a fixed support. Neglecting the lateral contraction, the elongation in the rod due to its own weight is $x \times 10^{-9}\text{ m}$. The value of $x$ is _______________.

(Given, young modulus Y = 2 $\times$ 1011 Nm−2 and g = 10 ms−2)

A wire of length L is hanging from a fixed support. The length changes to L1 and L2 when masses 1 kg and 2 kg are suspended respectively from its free end. Then the value of L is equal to:

In an experiment to determine the Young's modulus, steel wires of five different lengths $(1, 2, 3, 4$, and $5\text{ m})$ but of same cross section $\left(2\text{ mm}^2\right)$ 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 $x \times 10^{11}\text{ Nm}^{-2}$, then the value of $x$ is __________.

In an experiment to determine the Young's modulus of wire of a length exactly $1\text{ m}$, the extension in the length of the wire is measured as $0.4\text{ mm}$ with an uncertainty of $\pm 0.02\text{ mm}$ when a load of $1\text{ kg}$ is applied. The diameter of the wire is measured as $0.4\text{ mm}$ with an uncertainty of $\pm 0.01\text{ mm}$. The error in the measurement of Young's modulus $(\Delta \text{Y})$ is found to be $x \times 10^{10}\text{ Nm}^{-2}$. The value of $x$ is _________________. $($take $\text{g} = 10\text{ ms}^{-2}$)

Two wires of same length and radius are joined end to end and loaded. The Young's moduli of the materials of the two wires are Y1 and Y2. The combination behaves as a single wire then its Young's modulus is:

A uniform heavy rod of weight 10 kg ms−2, cross-sectional area 100 cm2 and length 20 cm is hanging from a fixed support. Young modulus of the material of the rod is 2 $\times$ 1011 Nm−2. Neglecting the lateral contraction, find the elongation of rod due to its own weight.

The length of a metal wire is l1, when the tension in it is T1 and is l2 when the tension is T2. The natural length of the wire is:

The value of tension in a long thin metal wire has been changed from T1 to T2. The lengths of the metal wire at two different values of tension T1 and T2 are l1 and l2 respectively. The actual length of the metal wire is:

The length of metallic wire is l1 when tension in it is T1. It is l2 when the tension is T2. The original length of the wire will be:

An object of mass m is suspended at the end of a massless wire of length L and area of crosssection A. Young modulus of the material of the wire is Y. If the mass is pulled down slightly its frequency of oscillation along the vertical direction is:

In an experiment, brass and steel wires of length 1 m each with areas of cross section 1mm2 are used. The wires are connected in series and one end of the combined wire is connected to a rigid support and other end is subjected to elongation. The stress required to produce a net elongation of 0.2 mm is, [Given, the Young's Modulus for steel and brass are, respectively, 120 × 109 N/m2 and 60 × 109 N/m2]

A load of mass M kg is suspended from a steel wire of length 2m and radius 1.0 mm in Searle's apparatus experiment. The increase in length produced in the wire is 4.0 mm. Now the load is fully immersed in a liquid of relative density 2. The relative density of the material of load is 8. The new value of increase in length of the steel wire is:

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 3 R, 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 thin 1 m long rod has a radius of 5 mm. A force of 50 $\pi$kN 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?

If the ratio of diameters, lengths and Young's modulus of steel and copper wires shown in the figure are p, q and s respectively, then the corresponding ratio of increase in their lengths would be

A copper wire of length 1.0 m and a steel wire of length 0.5 m having equal cross-sectional areas are joined end to end. The composite wire is stretched by a certain load which stretches the copper wire by 1 mm. If the Young's moduli of copper and steel are respectively $1.0 \times 10^{11}$ Nm ${}^{-2}$ and $2.0 \times 10^{11}$ Nm${}^{-2}$, the total extension of the composite wire is:

If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are a, b and c respectively, then the corresponding ratio of increase in their lengths is:

A wire elongates by $l\text{ mm}$ when a LOAD $W$ is hanged from it. If the wire goes over a pulley and two weights $W$ each are hung at the two ends, the elongation of the wire will be (in $\text{mm}$)