(Q1)A wire of cross- sectional area of 0.00006m2 and length 50cm stretches by 0.2mm under a load of 3000n.Calculate the Young's modulus for the wire (a)80000000000Nm-2(b)125000000000Nm-2(c)25000000000Nm-2(d)5000000000Nm-2 (Q2)a square plate of side 10cm is made of a metal of linear expansivity0.00002/k.As the plate is heated from 30c to 100c.The area of one face of the plate will increase to (a)1056cm2(b)104.2cm2(c)102.8cm2(d)101.4cm2
(Q1) Well, isn't it amazing how wires can stretch and still hold it together? It's like they're trying to do their yoga poses all the time! Anyway, let's calculate the Young's modulus for this wire.
We know that the cross-sectional area of the wire is 0.00006m2, the length is 50cm (or 0.5m), and it stretches by 0.2mm (or 0.0002m) under a load of 3000N.
Young's modulus (E) is defined as stress divided by strain. Stress is the force applied divided by the cross-sectional area, and strain is the change in length divided by the original length.
So let's do the math.
The stress is 3000N divided by 0.00006m2, which is 50,000,000N/m2.
The strain is 0.0002m divided by 0.5m, which is 0.0004.
Now, let's divide stress by strain to get the Young's modulus.
50,000,000N/m2 divided by 0.0004 equals 125,000,000,000N/m2.
So, the Young's modulus for this wire is option (b) 125,000,000,000Nm-2.
(Q2) Ah, the wonders of thermal expansion! So, we have a square plate with a side length of 10cm, made of a metal with a linear expansivity of 0.00002/k. And we need to figure out how its area changes as it heats up from 30°C to 100°C.
The formula for calculating the change in area due to temperature is:
ΔA = αAΔT
Where ΔA is the change in area, α is the linear expansivity, A is the original area, and ΔT is the change in temperature.
Let's plug in the values and find out the answer.
The original area is 10cm * 10cm, which is 100cm2.
The change in temperature is 100°C - 30°C, which is 70°C.
So, ΔA = 0.00002/k * 100cm2 * 70°C
Now let's do the math.
0.00002/k * 100cm2 * 70°C equals 0.14cm2.
Therefore, the area of one face of the plate will increase by 0.14cm2.
So, the new area will be 100cm2 + 0.14cm2, which is 100.14cm2.
Therefore, the area of one face of the plate will increase to option (d) 101.4cm2.
(Q1) To calculate the Young's modulus for the wire, we can use the formula:
Young's modulus (E) = (stress / strain)
where stress = force / area and strain = change in length / original length.
Given:
Cross-sectional area (A) = 0.00006 m^2
Length of wire (L) = 50 cm = 0.5 m
Extension in length (ΔL) = 0.2 mm = 0.0002 m
Force (F) = 3000 N
First, let's calculate the stress:
stress = force / area = 3000 N / 0.00006 m^2 = 50,000,000 N/m^2
Next, let's calculate the strain:
strain = ΔL / L = 0.0002 m / 0.5 m = 0.0004
Now, we can calculate the Young's modulus:
Young's modulus (E) = stress / strain = 50,000,000 N/m^2 / 0.0004 = 125,000,000,000 N/m^2 (option b)
Therefore, the correct answer is option (b) 125,000,000,000 N/m^2.
(Q2) To calculate the increase in area of the square plate, we can use the formula:
new area = original area * (1 + 2 * α * ΔT)
where original area = side length^2, α is the linear expansivity, and ΔT is the change in temperature.
Given:
Side length (L) = 10 cm = 0.1 m
Linear expansivity (α) = 0.00002 / K
Change in temperature (ΔT) = 100°C - 30°C = 70°C
First, let's calculate the original area:
original area = side length^2 = (0.1 m)^2 = 0.01 m^2
Next, let's calculate the new area:
new area = original area * (1 + 2 * α * ΔT)
= 0.01 m^2 * (1 + 2 * (0.00002 / K) * 70°C)
= 0.01 m^2 * (1 + 2 * 0.00002 * 70)
= 0.01 m^2 * (1 + 0.0028)
= 0.01 m^2 * 1.0028
= 0.010028 m^2
Converting the area back to cm^2:
new area = 0.010028 m^2 * (100 cm/m)^2
= 100.28 cm^2
Therefore, the area of one face of the plate will increase to approximately 100.28 cm^2, which is closest to option (c) 102.8 cm^2.
To answer Q1, we need to use Hooke's Law and Young's modulus formula.
Hooke's Law states that the strain of a material is directly proportional to the stress applied to it. Mathematically, it can be written as:
Stress = Young's modulus × Strain
In this case, the stress is the force applied divided by the cross-sectional area, and the strain is the change in length divided by the original length.
Given:
Cross-sectional area (A) = 0.00006 m²
Original length (L) = 50 cm = 0.5 m
Change in length (∆L) = 0.2 mm = 0.0002 m
Load (F) = 3000 N
To find Young's modulus (E), rearrange the formula:
E = Stress / Strain = (F / A) / (∆L / L)
Plug in the values:
E = (3000 N / 0.00006 m²) / (0.0002 m / 0.5 m)
E ≈ 125,000,000,000 N/m²
Therefore, the answer to Q1 is (b) 125,000,000,000 N/m².
To answer Q2, we need to use the formula for the change in area due to thermal expansion.
The change in area (∆A) can be calculated by multiplying the original area (A₀) by the coefficient of linear expansivity (α) and the change in temperature (∆T).
∆A = α × A₀ × ∆T
Given:
Side length (L) = 10 cm = 0.1 m
Coefficient of linear expansivity (α) = 0.00002/K
Change in temperature (∆T) = 100°C - 30°C = 70 K
To find the new area (A), add the change in area to the original area:
A = A₀ + ∆A = L² + α × A₀ × ∆T
Plug in the values:
A = (0.1 m)² + (0.00002/K) × (0.1 m)² × 70 K
A ≈ 0.01 m² + 0.000014 m²
A ≈ 0.010014 m²
To convert the area back to cm², multiply by 10,000:
A ≈ 0.010014 m² × 10,000 cm²/m²
A ≈ 100.14 cm²
Therefore, the answer to Q2 is (c) 102.8 cm².
Q1
σ=Eε
F/A = E ΔL/L
E= F•L/A• ΔL= 3000•0.5/0.00006•0.2•10⁻³ = 125000000000 N/m²
Q2
α= ΔL/L•ΔT
ΔL= α•L•ΔT=
=0.00002 •0.1•70 =0.00014 m
L₁=0.1+0.00014=0.10014 m
S₁=(0.10014)²=0.010028 m²=100.28 cm²