Advertisements
Advertisements
Question
Consider a long steel bar under a tensile stress due to forces F acting at the edges along the length of the bar (Figure). Consider a plane making an angle θ with the length. What are the tensile and shearing stresses on this plane
- For what angle is the tensile stress a maximum?
- For what angle is the shearing stress a maximum?
Advertisements
Solution
Consider the adjacent diagram,
Let the cross-sectional area of the bar be A. COnsider the equilibrium of the plane aa'. A force F must be acting on this plane making an angle `pi/2 - theta` with the normal ON. Resolving F into components, along the plane (FP) and normal to the plane.
`F_p = F cos θ`
`F_N = F sin θ`
Let the area of the face aa' be A, then
`A/A^' = sin θ`
∴ `A^' = A/sin θ`
The tensile stress = `"Normal force"/"Area" = (F sin θ)/A^'`
= `(F sin θ)/(A/sin θ)`
= `F/A sin^2 θ`
Shearing stress = `"Parallel force"/"Area"`
= `(F sin θ)/(A/sin θ)`
= `F/A sin θ. cos θ`
= `F/(2A) (2 sin θ. cos θ)`
= `F/(2A) sin 2θ`
a. For tensile stress to be maximum, `sin^2theta` = 1
⇒ `sin θ` = 1
⇒ θ = `pi/2`
b. For shearing stress to be maximum,
sin 2θ = 1
⇒ 2θ = `pi/2`
⇒ θ = `pi/4`
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 rod of length 1.05 m having negligible mass is supported at its ends by two wires of steel (wire A) and aluminium (wire B) of equal lengths as shown in Figure. The cross-sectional areas of wires A and B are 1.0 mm2 and 2.0 mm2, respectively. At what point along the rod should a mass m be suspended in order to produce (a) equal stresses and (b) equal strains in both steel and aluminium wires.
The yield point of a typical solid is about 1%. Suppose you are lying horizontally and two persons are pulling your hands and two persons are pulling your legs along your own length. How much will be the increase in your length if the strain is 1% ? Do you think your yield point is 1% or much less than that?
The breaking stress of a wire depends on
A wire can sustain the weight of 20 kg before breaking. If the wire is cut into two equal parts, each part can sustain a weight of
A heave uniform rod is hanging vertically form a fixed support. It is stretched by its won weight. The diameter of the rod is
Answer in one sentence.
How should be a force applied on a body to produce shearing stress?
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 ______.
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 wire is suspended from the ceiling and stretched under the action of a weight F suspended from its other end. The force exerted by the ceiling on it is equal and opposite to the weight.
- Tensile stress at any cross section A of the wire is F/A.
- Tensile stress at any cross section is zero.
- Tensile stress at any cross section A of the wire is 2F/A.
- Tension at any cross section A of the wire is F.
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 ______.
If 'S' is stress and 'Y' is young's modulus of the material of a wire, the energy stored in the wire per unit volume is ______.
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 ______.
The area of the cross-section of the rope used to lift a load by a crane is 2.5 × 10-4m2. 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)
Answer in one sentence:
What is an elastomer?
