a metal wire 1.0 mm in diameter and 2.0 m long hangs vertically with a 5.0 kg mass suspended from it. if the wire stretches 1.2 mm under the tension, what is the value of the young's modulus for metal?
Not correct
To calculate the value of the Young's modulus for the metal wire, we can use the formula:
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.
To find the value of Young's modulus for the metal, we need to use Hooke's Law, which relates the stress applied to an object to its strain.
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.
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