Stress = E*strain
E is Young's modulus
E*(delta L/L) = M*g/(pi*R^2)
deltaL/L - 1.2/2000 = 6*10^-4
is the strain
M*g/(pi*R^2) = 6.2*10^7 N/m^2
Solve for E. It is about 10^11 N/m^2
E is Young's modulus
E*(delta L/L) = M*g/(pi*R^2)
deltaL/L - 1.2/2000 = 6*10^-4
is the strain
M*g/(pi*R^2) = 6.2*10^7 N/m^2
Solve for E. It is about 10^11 N/m^2
Young's modulus (Y) = (F * L) / (A * ΔL),
where:
- F is the force exerted on the wire (weight of the mass),
- L is the original length of the wire,
- A is the cross-sectional area of the wire, and
- ΔL is the change in length of the wire.
Let's substitute the given values into the formula:
F = 5.0 kg * 9.8 m/s²
(considering acceleration due to gravity as 9.8 m/s², assuming Earth's surface)
F = 49 N
L = 2.0 m
A = π * (d/2)²
= π * (1.0 mm / 2)²
= π * (0.5 mm)²
= π * 0.25 mm²
≈ 0.785 mm²
(converting the diameter to radius and using the formula for the area of a circle)
ΔL = 1.2 mm
Now, we can substitute the values into the formula and calculate the Young's modulus:
Y = (F * L) / (A * ΔL)
= (49 N * 2.0 m) / (0.785 mm² * 1.2 mm)
First, let's convert the units to be consistent:
1 mm = 1 × 10^(-3) m,
1 mm² = (1 × 10^(-3) m)² = 1 × 10^(-6) m²
Y = (49 N * 2.0 m) / (0.785 × 10^(-6) m² * 1.2 × 10^(-3) m)
= (98 N * m) / (0.942 × 10^(-9) m⁴)
= (98 N * m) / 9.42 × 10^(-10) N*m²
Simplifying by dividing numerator and denominator by 10^(-9):
Y = (98 N * m) / (0.942 * 10)
= 98 N * m / 9.42
≈ 10.4 × 10⁹ N/m²
= 10.4 GPa
Thus, the value of the Young's modulus for the metal wire is approximately 10.4 GPa.
Hooke's Law states that within the elastic limit of a material, the stress applied is directly proportional to the strain produced. Mathematically, it can be expressed as:
Stress (σ) = Young's Modulus (Y) * Strain (ε)
Where:
σ = Stress
Y = Young's Modulus
ε = Strain
In this case, we need to find Young's Modulus (Y). We are given the following information:
Diameter of the wire (d) = 1.0 mm = 0.001 m
Radius of the wire (r) = d/2 = 0.001/2 = 0.0005 m
Length of the wire (L) = 2.0 m
Mass suspended (m) = 5.0 kg
Wire stretch (δL) = 1.2 mm = 0.0012 m
To calculate the strain (ε), we use the formula:
Strain (ε) = Wire Stretch (δL) / Original Length (L)
ε = 0.0012 m / 2.0 m
ε = 0.0006
Now, rearranging Hooke's Law equation, we get:
Young's Modulus (Y) = Stress (σ) / Strain (ε)
We need to calculate the stress (σ) first. Stress is defined as:
Stress (σ) = Force (F) / Area (A)
The force in this case is the weight of the mass suspended, which is given by:
Force (F) = mass (m) * gravity (g)
F = 5.0 kg * 9.8 m/s²
F = 49 N
The area can be calculated using the formula for the cross-sectional area of a wire:
Area (A) = π * Radius²
A = π * (0.0005 m)²
A = 3.14 * 0.00000025 m²
A = 7.85 x 10^-7 m²
Now we can calculate the stress:
Stress (σ) = 49 N / 7.85 x 10^-7 m²
σ ≈ 6.24 x 10^7 N/m²
Finally, substituting the values into the Young's Modulus equation:
Young's Modulus (Y) = 6.24 x 10^7 N/m² / 0.0006
Y ≈ 1.04 x 10^11 N/m²
Therefore, the value of Young's Modulus for the metal is approximately 1.04 x 10^11 N/m².