Advertisements
Advertisements
प्रश्न
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.)
Advertisements
उत्तर
Consider an element of width dr at t as shown in the diagram.
Let T(r) and T(r + dr) be the tensions at r and r + dr respectively.
Net centrifugal force on the element = ω2rdm ....(Where ω is the angular velocity of the rod)
= ω2rμdr .....(∵ μ = mass/length)
⇒ T(r) – T(r + dr) = μω2rdr
⇒ – dT = μω2rdr ......[∵ Tension and centrifugal forces are opposite]
∴ `- int_(T = 0)^T dT = int_(r = l)^(r= r) μω^2rdr` ......[∵ T = 0 at r = l]
⇒ `T(r) = (μω^2)/2 (l^2 - r^2)`
Let the increase in length of the element dr be Δr
So, Young's modulus Y = `"Stress"/"Strain" = ((T(r))/A)/((Δr)/(dr))`
∴ `(Δr)/(dr) = (T(r))/A = (μω^2)/(2YA) (l^2 - r^2)`
∴ `Δr = 1/(YA) (μω^2)/2 (l^2 - r^2)dr`
∴ Δ = Change in length in right part = `1/(YA) (μω^2)/2 int_0^l (l^2 - r^2) dr`
= `(1/(YA)) (μω^2)/2 [t^3 - l^3/3]`
= `1/(3YA) μω^2l^2`
∴ Total change in length = 2Δ = `2/(3YA) μω^2l^2`
APPEARS IN
संबंधित प्रश्न
Four identical hollow cylindrical columns of mild steel support a big structure of mass 50,000 kg. The inner and outer radii of each column are 30 cm and 60 cm respectively. Assuming the load distribution to be uniform, calculate the compressional strain of each column.
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
The length of a metal wire is l1 when the tension in it T1 and is l2 when the tension is T2. The natural length of the wire is
Consider the situation shown in figure. The force F is equal to the m2 g/2. If the area of cross section of the string is A and its Young modulus Y, find the strain developed in it. The string is light and there is no friction anywhere.
A copper wire of cross-sectional area 0.01 cm2 is under a tension of 20N. Find the decrease in the cross-sectional area. Young modulus of copper = 1.1 × 1011 N m−2 and Poisson ratio = 0.32.
`["Hint" : (Delta"A")/"A"=2(Delta"r")/"r"]`
A uniform rectangular block of mass of 50 kg is hung horizontally with the help of three wires A, B and C each of length and area of 2m and 10mm2 respectively as shown in the figure. The central wire is passing through the centre of gravity and is made of material of Young's modulus 7.5 x 1010 Nm−2 and the other two wires A and C symmetrically placed on either side of the wire B are of Young's modulus 1011 Nm−2 The tension in the wires A and B will be in the ratio of:
The temperature of a wire is doubled. The Young’s modulus of elasticity ______.
A rigid bar of mass M is supported symmetrically by three wires each of length l. Those at each end are of copper and the middle one is of iron. The ratio of their diameters, if each is to have the same tension, is equal to ______.
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).
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?
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 uniform metal rod of 2 mm2 cross section is heated from 0°C to 20°C. The coefficient of linear expansion of the rod is 12 × 10-6/°C, it's Young's modulus is 1011 N/m2. The energy stored per unit volume of the rod is ______.
Young's modulus is also known as
What does the dimensional formula [L⁻¹M¹T⁻²] represent?
Which of the following statements about Young's modulus is correct?
