20 Physics -- Elasticity

What stress would cause a wire to increase in length by one-tenth of one percent if the Young modulus for the wire is 12 × 1010 N/m2 ? What force would produce this stress if the diameter of the wire is 0.56 mm?

What stress would cause a wire to increase in length by one-tenth of one percent if the Young modulus for the wire is 12 × 10¹⁰ N/m² ? What force would produce this stress if the diameter of the wire is 0.56 mm?

1 Answers

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Solution:
The Young modulus of a wire, Y = 12 × 1010 N/m2 ;
the diameter of the wire, d = 0.56 mm = 0.56 × 10^–3 mm
The cross-section area of the wire,

The strain = one-tenth of 1 percent

In order to oscillate at 10Hz, a mass of 20g should be suspended from a spring of spring constanta. 2.5N/mb. 12.6N/mc. 79N/md. 9.7N/m

If the oscillation is simple harmonic motion then we have,

Inserting given data, we get the spring constant approximately equal to 79N/m.

Particle 1 experiences a perfectly elastic collision with a stationary particle 2. Determine their mass ratio if the particles fly apart symmetrically relative to the initial motion direction of pa...

When they fly apart symmetrically relative to the initial motion direction with the angle of divergence,
From the conservation of momentum, along the horizontal and vertical direction

and m1v1sin(θ/2)=m2v2sin(θ/2)

or, m1v1=m2v2 ...(2)

Now, from conservation of kinetic energy,

1/2 m1 v1²+ 0 =1/2 m1 v1²+1/2 m2 v2² ...(3)

From (1) and (2),

m1 u1=cos(θ/2)(m1v1+m1v1m2m2)=2m1v1cos(θ/2)

So, u1=2v1cos(θ/2)...(4)

From (2), (3) and (4)

4m1cos²(θ/2) v1²=m1v1²+m2m1²v1²/m2²

or, 4cos²(θ/2)=1+m1/m2

or,...

The material in human bones and elephant bones essentially the same, but an elephant has thicker legs. Explain why, in terms of breaking stress.

Breaking stress may occur when the material is excessively stretched or compressed. To withstand the heavy weight of an elephant, it has thicker legs so that the ratio of the weight to the cross-section area of the legs does not exceed the breaking stress of the bone material. To withstand the weight of a man, he can have thin legs and the ratio of his weight to the cross-section area of the leg will not exceed the breaking stress for the bone material. This is why an elephant has thicker...

Explain which one is more elastic–rubber or steel.

Suppose two wires of steel and rubber of equal length L and cross-section A are stretched by the same force F. The Young modulus for steel is steel steel and similarly that for rubber iswhere l_steelis the extension of the steel wire and l_rubberis that of the rubber wire. Thus taking the ratio, we have

As l_rubber> l_steel, Y_steel> Y_rubber. The more the value of the elastic constant, the more the object is elastic because a very large force or stress is needed to make it break. This means...

A solid sphere of radius 10 cm is subjected to a uniform pressure equal to 5 × 108 N/m2 . The bulk modulus of the material of the sphere is 3.14 × 1011 N/m2 . Find the change in its volume.

Solution:
The radius of a sphere, r = 10 cm = 0.10 m;
the increase in pressure, p = 5 10⁸N/m²;
the bulk modulus, B = 3.14 10¹¹N/m²;
the change in volume, V = ?

An aluminum cube of each side 4 cm is subjected to a tangential force. The top face of the cube is sheared 0.012 cm w.r.t. the bottom. Find (i) shearing strain (ii) shear stress and (iii) shearing ...

Solution:
The side of a cube, L = 4 cm = 0.04 m;
the shear length, x = 0.012 cm = 1.2 10^–4m

Find the extension produced in a copper wire of length 2 m and diameter of 3 mm when a force of 30 N is applied. (Young's modulus for copper = 1.1 × 1011 N/m2 ).

Solution:
The extension, l = ?;
the force applied, F = 30 N;
the length of a copper wire, L = 2 m;
the diameter of the wire, d = 3 mm = 3 10^–3m
The cross-section area of the wire,

A cylindrical of copper wire and a cylindrical steel wire, each of length 1.5 m and diameter 2 mm, are joined at one end to form a composite wire 3 m long. The wire is loaded until its length becom...

Solution:
The length of a copper wire, LC = 1.5 m;
the length of a steel wire, LS = 1.5 m;
the diameter of the wires, d = 2 mm = 2 10^–3m;
the composite length, L = 3 m;
the extended composite length, L' = 3.003 m;
the total extension, l = L' – L = 3.003 – 3 = 0.003 m;
the extension of the copper wire, l_C= ?;
the extension of the steel wire, l_S= ?

l_C+ l_S= 0.003 … (1)

For the copper wire, Young modulus,

A uniform steel wire of density 7800 kg/m3 weighs 16 g and is 250 cm long. It lengthens by 1.2 mm when stretched by a force of 80 N. Calculate (i) the value of Young's modulus for steel, (ii) the e...

Calculate the longest length of a steel wire that can hang vertically without breaking. Breaking stress for steel = 7.982 × 108 N/m2 and density for steel = 8.1 × 103 kg/m3 . (g = 9.8 m/s2 ).

Solution:
The breaking stress = 7.982 10⁸N/m²;
the density of steel,ρ= 8.1 10³kg/m³
Suppose A is the area of cross-section of the steel wire and L is the longest length of the steel wire.
We have,
the weight of the wire = mg = (volume density)g
or F = LAρg

A copper rod of length L is suspended from the ceiling by one of its ends. Find the elongation ΔL of the rod due to its own weight.

Solution:

Let m be the mass of the copper rod, g be the acceleration due to gravity and A be the cross-sectional area. It is given that L is the length of the rod then assume e to be the elongation produced. If Y be the Young's modulus of elasticity of copper, then

Y = FL/Ae

e = FL/AY

Further, F = mg and centre of mass of the rod is at the centre, so L' = L/2

Therefore, e = mgL/2AY

Steel is more elastic than rubber. Give your opinion.

The strain produced in rubber is much larger compared to that in steel. This means that steel has a larger value of Young's modulus of elasticity and hence, steel has more elasticity than rubber.

Why are rubbers used as vibration absorbers?

Rubber is used as vibration absorbers, because rubber has a relatively high shear modulus compared to other materials. That means when a rubber material is shear stressed, i.e. stressed parallel to its cross-section, rubber can be stressed more before it becomes permanently deformed.

Why does a wire get heated when it is bent back and fourth?

When a wire is bent back and forth, heat is generated due to the area of the elastic hysteresis and frictional force. Hence it becomes hot.

Explain why soldiers are ordered to break steps while crossing a bridge.

If they will not break their steps, the frequency of marching may coincide with natural frequency of the bridge. Due to this, bridge vibrates with maximum amplitude. If the amplitude of vibration exceeds elastic limit then bridge will collapse. So to avoid this, marching soldiers are ordered to break their steps.

Note of chapter 8

Does the temperature affect elasticity?

As the temperature increases, the interatomic forces of attraction become weaker. For a given stress, a large strain is produced when temperature increased.

Hence, the modulus of elasticity(stress/strain) decreases with increasing temperature.

So YES, temperature does affect the elasticity of the body, i.e body becomes more plastic in nature at higher temperature.

The stress-strain graphs for materials A and B are shown in Figure The graphs are drawn to the same scale. (a) Which of the materials has the greater Young’s modulus?(b) Which of the material is mo...

(a)
Young’s modulus is given by the equation,Y= stress/ strainAt a particular strain, Young’s modulus is directly proportional to the stress.
The slope of stress strain curve for material A has greater than the material B which gives higher value of Young's modulus.

(b)Material B is more brittle since the fracture point (end of solid curve) is at lower corresponding value of stress.

(d)The...

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