A steel wire 2.7-mm in diameter stretches by 0.211 percent when a mass is suspended from it. How large is the mass in kg?

wondering what temperature this occurs at.

dl/l=(force/area)/young's modulus

force=mass*g=youngMod*area*dl/l
dl/l= .0211
area=PI(.0027/2)^2
g= 9.8N/kg
mass(in kg)=youngMod*area*dl/l*1/9.8

Young mod is about 200E9 N/m^2 about, but it depends on temperature.

To calculate the mass in kg, we need to know the density of the steel wire. The density of steel is typically around 7850 kg/m³.

Step 1: Calculate the cross-sectional area of the wire
The cross-sectional area of a wire can be calculated using the formula:
Area = π * (diameter/2)²
where π is approximately 3.14159 and the diameter is given as 2.7 mm.

Area = 3.14159 * (2.7 mm/2)²
Area = 3.14159 * (1.35 mm)²
Area = 3.14159 * 1.8225 mm²
Area = 5.72478 mm²

Step 2: Convert the cross-sectional area from mm² to m²
To calculate the mass, we need to have the cross-sectional area in m². Since 1 m = 1000 mm, we need to divide the cross-sectional area by 1,000,000.

Area = 5.72478 mm² / 1,000,000
Area = 0.00000572478 m²

Step 3: Calculate the change in length of the wire
The change in length can be calculated using the equation:
Change in length = (strain) * (original length)
The strain is given as 0.211 percent and the original length is unknown.

Step 4: Calculate the original length
Let's assume the original length of the wire is L.

Change in length = (0.211/100) * L
Change in length = 0.00211 * L

Step 5: Calculate the mass
The mass can be calculated using the equation:
Mass = (force) / (acceleration due to gravity)
The force can be calculated using Hooke's Law:
force = (stress) * (area)
The stress can be calculated using the equation:
Stress = (strain) * (Young's modulus)

The Young's modulus of steel is typically around 200 GPa (gigapascal), which is equivalent to 200,000,000 N/m².

Stress = (0.00211) * (200,000,000 N/m²)
Stress = 422,000 N/m²

force = (422,000 N/m²) * (0.00000572478 m²)
force = 2.41865 N

Mass = (2.41865 N) / (9.8 m/s²)
Mass = 0.24707 kg

Therefore, the mass is approximately 0.24707 kg.

To find the mass, we first need to understand the relationship between the tension in the wire and the amount it stretches. This can be determined by Hooke's Law, which states that the force applied to an object is directly proportional to its extension or deformation.

The formula for Hooke's Law is: F = k * ΔL

Where:
F is the force applied to the wire,
k is the spring constant (a measure of the wire's stiffness),
ΔL is the change in length of the wire.

In this case, the change in length is given as a percentage (0.211%) of the original length. To convert it to a decimal, divide the percentage by 100:
ΔL = (0.211/100) * original_length

Now, let's consider the relationship between the force applied to the wire and the mass suspended from it. The force applied to the wire is equal to the weight of the hanging mass, which can be calculated using the formula: F = m * g

Where:
m is the mass of the object,
g is the acceleration due to gravity (approximately 9.8 m/s²).

Now we can combine both equations:

F = k * ΔL
m * g = k * ΔL

Since we are trying to find the mass (m), we can rearrange the equation:

m = (k * ΔL) / g

To determine the value of the spring constant (k), we need to know the material and specific properties of the wire. Let's assume a typical value for a steel wire, which is 2 * 10^11 N/m².

Now we can substitute the values into the equation and solve for the mass (m):

m = (2 * 10^11 * iven that the diameter of the wire is 2.7 mm, we can calculate its radius (r) by dividing the diameter by 2 (since the diameter is twice the radius):

r = 2.7 mm / 2 = 1.35 mm = 0.00135 m

Next, we need to find the original length (L) of the wire based on its diameter (d). We can use the formula for the circumference of a circle:

C = π * d

Rearranging the formula to find the length (L):

L = C / π

Substituting the values:

L = (π * d) / π = d

Therefore, the original length (L) of the wire is equal to its diameter (d) since the wire was assumed to be a straight line before stretching.

Now we can calculate the change in length (ΔL):

ΔL = (0.211/100) * L = (0.211/100) * d

Finally, we can substitute the values into the mass formula:

m = (2 * 10^11 * (0.211/100) * d) / g

Given that the diameter (d) is 2.7 mm and the acceleration due to gravity (g) is 9.8 m/s², we can calculate the mass (m).