To solve this problem, we need to use the following formulas:
1. Stress (σ) = Force (F) / Area (A)
2. Strain (ε) = Change in Length (ΔL) / Original Length (L)
3. Young's Modulus (E) = Stress (σ) / Strain (ε)
4. Spring Constant (k) = Force (F) / Extension (x)
First, we need to find the force exerted by the copper on the spring wire. We can use the formula:
Force (F) = mass (m) x gravity (g)
where mass (m) = 345 g = 0.345 kg, and gravity (g) = 9.81 m/s^2.
Therefore, F = 0.345 kg x 9.81 m/s^2 = 3.38 N.
1.1. The stress of the spring is:
σ = F / A
where A is the area of the spring wire. The diameter of the spring wire is 30 cm, which gives a radius of 15 cm or 0.15 m. Therefore, the area of the wire is:
A = πr^2 = π(0.15 m)^2 = 0.0707 m^2.
Thus, the stress of the spring is:
σ = 3.38 N / 0.0707 m^2 = 47.8 N/m^2.
1.2. The strain of the spring is:
ε = ΔL / L
where ΔL is the change in length of the spring and L is the original length of the spring. The spring stretches from 75 cm to 80 cm, which gives a change in length of:
ΔL = 80 cm - 75 cm = 0.05 m.
The original length of the spring is half the distance it stretches when holding the copper, or:
L = (75 cm + 80 cm) / 2 = 77.5 cm = 0.775 m.
Thus, the strain of the spring is:
ε = 0.05 m / 0.775 m = 0.0645.
1.3 The Young's modulus of the spring is defined as the ratio of stress to strain. We have already calculated the stress and strain of the spring in the previous parts of the question. So the Young's modulus of the spring is:
Young's modulus (E) = stress (σ) / strain (ε)
E = 47.8 N/m^2 / 0.0645 = 740.9 N/m^2
Therefore, the Young's modulus of the spring is 740.9 N/m^2.
1.4 The spring constant is defined as the ratio of the force applied to the spring to the resulting extension. We have already calculated the force applied to the spring in the previous parts of the question. To calculate the spring constant, we need to find the extension of the spring.
The extension of the spring is the difference between the final length and the original length. The original length of the spring is half the distance it stretches when holding the copper, or:
Original length (L) = (75 cm + 80 cm) / 2 = 77.5 cm = 0.775 m
The final length of the spring when holding the copper is 80 cm. So the extension of the spring is:
Extension (x) = Final length - Original length
x = 80 cm - 77.5 cm = 2.5 cm = 0.025 m
Therefore, the spring constant is:
Spring constant (k) = Force (F) / Extension (x)
k = 3.38 N / 0.025 m = 135.2 N/m
Therefore, the spring constant is 135.2 N/m.