A 3.8 kg object is supported by an aluminum wire of length 1.9 m and diameter 2.2 mm. How much will the wire stretch?
To determine how much the wire will stretch, we need to consider the weight of the object and the properties of the wire material.
The first step is to calculate the force due to the weight of the object. We can use the formula:
Force (F) = mass (m) * acceleration due to gravity (g)
Given that the mass (m) of the object is 3.8 kg and the acceleration due to gravity (g) is approximately 9.8 m/s^2, we can calculate the force:
F = 3.8 kg * 9.8 m/s^2
Next, we need to determine the cross-sectional area of the wire. The formula to calculate the cross-sectional area (A) is:
A = π * r^2
where π is a mathematical constant (approximately 3.14) and r is the radius of the wire.
The diameter given is 2.2 mm, which means the radius (r) is half of that:
r = (2.2 mm) / 2 = 1.1 mm = 0.0011 m
Using the radius, we can calculate the cross-sectional area:
A = π * (0.0011 m)^2
Now we can determine the stress on the wire. Stress (σ) is calculated using the formula:
σ = F / A
where F is the force and A is the cross-sectional area.
Given the calculated force and cross-sectional area, we can compute the stress:
σ = F / A
Finally, we can use Hooke's law to determine how much the wire will stretch. Hooke's law states that the extension of an elastic material is directly proportional to the applied force. The formula is:
ΔL = (F * L) / (E * A)
where ΔL is the change in length, F is the force, L is the original length of the wire, E is the modulus of elasticity of aluminum, and A is the cross-sectional area.
The modulus of elasticity of aluminum is approximately 70 GPa (gigapascals), which is equivalent to 70 * 10^9 Pa.
Given the length of the wire is 1.9 m, we can substitute the values into the formula to calculate the change in length:
ΔL = (F * L) / (E * A)
Now, let's plug in the values into the equation:
ΔL = (F * L) / (E * A)
Note: Calculations using specific unit conversions (such as converting mm to m) have been omitted to simplify the explanation.
Once you substitute the calculated values into the equation, you should be able to determine the amount by which the aluminum wire will stretch under the given conditions.