Advertisements
Advertisements
Question
A steel wire of mass µ per unit length with a circular cross section has a radius of 0.1 cm. The wire is of length 10 m when measured lying horizontal, and hangs from a hook on the wall. A mass of 25 kg is hung from the free end of the wire. Assuming the wire to be uniform and lateral strains << longitudinal strains, find the extension in the length of the wire. The density of steel is 7860 kg m–3 (Young’s modules Y = 2 × 1011 Nm–2).
Advertisements
Solution
Consider the diagram when a small element of length dx is considered at x from the load (x = 0).
Let T(x) and T(x + dx) are tensions on the two cross-sections a distance dx apart, then – t(x + dx) + T(x) = dmg = μ dxg (where μ is the mass/length) ......(∵ dm = μdx)
dT = μgdx .....[∵ dT = T(x + dx) – T(x)]
⇒ T(x) = μgx + C .....(On integrating)
At x = 0, T(0) = Mg
⇒ C = mg
∴ T(x) = μgx + Mg
Let the length dx at x increase by dr, then
Young's modulus Y = `"Stress"/"Strain"`
`((T(x))/A)/((dr)/(dx)) = Y`
⇒ `(dr)/(dx) = 1/(YA) T(x)`
⇒ `r = 1/(YA) int_0^L (μgx + Mg)dx`
= `1/(YA) [(μgx^2)/2 + Mgx]_0^L`
= `1/(YA)[(mgL^2)/2 + MgL]` ......(m is the mass of the wire)
`A = pi xx (10^-3)^2 m^2`
`Y = 200 xx 10^9 Nm^-2`
`m = pi xx (10^-3)^2 xx 10 xx 7860` kg
∴ `r = 1/(2 xx 10^11 xx pi xx 10^-6)` ......`[(pi xx 786 xx 10^-3 xx 10 xx 10)/2 + 25 xx 10 xx 10]`
= `[196.5 xx 10^-6 + 3.98 xx 10^-3]`
= 4 × 10–3 m
APPEARS IN
RELATED QUESTIONS
The figure shows the strain-stress curve for a given material. What are (a) Young’s modulus and (b) approximate yield strength for this material?
Two wires of diameter 0.25 cm, one made of steel and the other made of brass are loaded as shown in Fig. 9.13. The unloaded length of steel wire is 1.5 m and that of brass wire is 1.0 m. Compute the elongations of the steel and the brass wires.
Two wires A and B are made of same material. The wire A has a length l and diameter rwhile the wire B has a length 2l and diameter r/2. If the two wires are stretched by the same force, the elongation in A divided by the elongation in B is
A wire elongates by 1.0 mm when a load W is hung from it. If this wire goes over a a pulley and two weights W each are hung at the two ends, he elongation of he wire will be
A student plots a graph from his reading on the determination of Young modulus of a metal wire but forgets to put the labels. the quantities on X and Y-axes may be respectively
(a) weight hung and length increased
(b) stress applied and length increased
(c) stress applied and strain developed
(d) length increased and the weight hung.
Young's modulus of a perfectly rigid body is ______.
The temperature of a wire is doubled. The Young’s modulus of elasticity ______.
The Young’s modulus for steel is much more than that for rubber. For the same longitudinal strain, which one will have greater tensile stress?
A truck is pulling a car out of a ditch by means of a steel cable that is 9.1 m long and has a radius of 5 mm. When the car just begins to move, the tension in the cable is 800 N. How much has the cable stretched? (Young’s modulus for steel is 2 × 1011 Nm–2.)
If the yield strength of steel is 2.5 × 108 Nm–2, what is the maximum weight that can be hung at the lower end of the wire?
A steel rod of length 2l, cross sectional area A and mass M is set rotating in a horizontal plane about an axis passing through the centre. If Y is the Young’s modulus for steel, find the extension in the length of the rod. (Assume the rod is uniform.)
A boy's catapult is made of rubber cord which is 42 cm long, with a 6 mm diameter of cross-section and negligible mass. The boy keeps a stone weighing 0.02 kg on it and stretches the cord by 20 cm by applying a constant force. When released, the stone flies off with a velocity of 20 ms-1. Neglect the change in the area of the cross-section of the cord while stretched. Young's modulus of rubber is closest to ______.
If the length of a wire is made double and the radius is halved of its respective values. Then, Young's modules of the material of the wire will ______.
The force required to stretch a wire of cross section 1 cm2 to double its length will be ______.
(Given Young's modulus of the wire = 2 × 1011 N/m2)
Young's modulus is also known as ______.
What does the dimensional formula [L⁻¹M¹T⁻²] represent?
What is longitudinal strain?
The maximum elongation of a steel wire of 1 m length if the elastic limit of steel and its Young’s modulus, respectively, are 8 × 108 Nm−2 and 2 × 1011 Nm−2, is ______.
