Advertisements
Advertisements
Question
A rod of length l and negligible mass is suspended at its two ends by two wires of steel (wire A) and aluminium (wire B) of equal lengths (Figure). The cross-sectional areas of wires A and B are 1.0 mm2 and 2.0 mm2, respectively.
(YAl = 70 × 109 Nm−2 and Ysteel = 200 × 109 Nm–2)
- Mass m should be suspended close to wire A to have equal stresses in both the wires.
- Mass m should be suspended close to B to have equal stresses in both the wires.
- Mass m should be suspended at the middle of the wires to have equal stresses in both the wires.
- Mass m should be suspended close to wire A to have equal strain in both wires.
Advertisements
Solution
b and d
Explanation:
Let the mass is placed at x from the end B.
Let TA and TB be the tensions in wire A and wire B respectively.
For the rotational equilibrium of the system,
∑τ = 0 ......(Total torque = 0)
⇒ `T_Bx = T_A (l - x)` = 0
⇒ `T_B/T_A = (l - x)/x` ......(i)
Stress in wire `A = S_A = T_A/a_A`
Stress in wire `B = S_B = T_B/a_B`
Where aA and aB are cross-sectional areas of wire A and B respectively.
By question aB = 2aA
Now, for equal stress SA = SB
⇒ `T_A/a_A = T_B/a_B`
⇒ `T_B/T_A = a_B/a_A` = 2
⇒ `(l - x)/x` = 2
⇒ `l/x - 1` = 2
⇒ `x = l/3`
⇒ `l - x = l - l/3 = (2l)/3`
Hence, mass m should be placed closer to B.
For equal train, (strain)A = (strain)B
⇒ `(Y_A)/S_A = Y_B/S_B` ......(Where YA and YB are Young moduli)
⇒ `Y_(steel)/(T_A/a_A) = Y_(Al)/(T_B/a_B)`
⇒ `Y_(steel)/Y_(Al) = T_A/T_B xx a_B/a_A = (x/(l - x))((2a_A)/a_A)`
⇒ `(200 xx 10^9)/(70 xx 10^9) = (2x)/(l - x)`
⇒ `20/7 = (2x)/(l - x)`
⇒ `10/7 = x/(l - x)`
⇒ `10l - 10x = 7x`
⇒ `17x = 10l`
⇒ `x = (10l)/17`
`l - x = l - (10l)/17 = (7l)/17`
Hence, mass m should be placed closer to wire A.
APPEARS IN
RELATED QUESTIONS
A steel cable with a radius of 1.5 cm supports a chairlift at a ski area. If the maximum stress is not to exceed 108 N m–2, what is the maximum load the cable can support?
A rigid bar of mass 15 kg is supported symmetrically by three wires each 2.0 m long. Those at each end are of copper and the middle one is of iron. Determine the ratio of their diameters if each is to have the same tension.
Determine the volume contraction of a solid copper cube, 10 cm on an edge, when subjected to a hydraulic pressure of 7.0 ×106 Pa.
A mild steel wire of length 1.0 m and cross-sectional area 0.50 × 10–2 cm2 is stretched, well within its elastic limit, horizontally between two pillars. A mass of 100 g is suspended from the mid-point of the wire. Calculate the depression at the midpoint.
Two strips of metal are riveted together at their ends by four rivets, each of diameter 6.0 mm. What is the maximum tension that can be exerted by the riveted strip if the shearing stress on the rivet is not to exceed 6.9 × 107 Pa? Assume that each rivet is to carry one-quarter of the load.
A rope 1 cm in diameter breaks if the tension in it exceeds 500 N. The maximum tension that may be given to a similar rope of diameter 2 cm is
A load of 10 kg is suspended by a metal wire 3 m long and having a cross-sectional area 4 mm2. Find (a) the stress (b) the strain and (c) the elongation. Young modulus of the metal is 2.0 × 1011 N m−2.
A charged particle is moving in a uniform magnetic field in a circular path of radius R. When the energy of the particle becomes three times the original, the new radius will be ______.
A spiral spring is stretched by a weight. The strain will be:
The maximum load a wire can withstand without breaking, when its length is reduced to half of its original length, will ______.
A spring is stretched by applying a load to its free end. The strain produced in the spring is ______.
A mild steel wire of length 2L and cross-sectional area A is stretched, well within elastic limit, horizontally between two pillars (Figure). A mass m is suspended from the mid point of the wire. Strain in the wire is ______.
A rectangular frame is to be suspended symmetrically by two strings of equal length on two supports (Figure). It can be done in one of the following three ways;
| (a) |  |
| (b) |  |
| (c) |  |
The tension in the strings will be ______.
Consider two cylindrical rods of identical dimensions, one of rubber and the other of steel. Both the rods are fixed rigidly at one end to the roof. A mass M is attached to each of the free ends at the centre of the rods.
A steel wire having a radius of 2.0 mm, carrying a load of 4 kg, is hanging from a ceiling. Given that g = 3.1πms-2, what will be the tensile stress that would be developed in the wire?
A body of mass m = 10 kg is attached to one end of a wire of length 0.3 m. The maximum angular speed (in rad s-1) with which it can be rotated about its other end in the space station is (Breaking stress of wire = 4.8 × 107 Nm-2 and the area of cross-section of the wire = 10-2 cm2) is ______.
What is an elastomer?
Strain is defined as the ratio of:
The SI unit of stress is:
What are the dimensions of strain?
