A steel guitar string is 73 cm long and 0.15 mm in diameter. If it's under tension of 1.5kN, how much is it stretched?
Stress = E*Strain = E*(deltaL/L)
For steel, E = 200*10^9 N/m^2
E is called Young's modulus
Calculate the stress in N/m^2 and solve for deltaL (the length change) in meters
To determine how much a steel guitar string is stretched under tension, you can use Hooke's Law. Hooke's Law states that the amount of elongation or compression of an object is directly proportional to the force applied to it, as long as the material behaves linearly (does not exceed its elastic limit).
The formula for Hooke's Law is:
F = k * x
Where:
F is the force applied (in newtons)
k is the spring constant (in newtons per meter)
x is the amount of elongation or compression (in meters)
To find the amount of elongation (x) of the guitar string, we need to rearrange the formula:
x = F / k
In this case, the force applied (F) is given as 1.5 kN (kilonewtons). We need to convert it to newtons by multiplying it by 1000:
F = 1.5 kN * 1000 = 1500 N
Now, we need to find the spring constant (k). The spring constant depends on the material properties, such as the Young's modulus, cross-sectional area, and length of the guitar string. In this case, the guitar string is made of steel.
The spring constant (k) can be calculated using Young's modulus (E) and the cross-sectional area (A) of the string:
k = A * E / L
Where:
A is the cross-sectional area of the string (in square meters)
E is the Young's modulus of steel (in pascals)
L is the length of the string (in meters)
To find the cross-sectional area (A), we need to use the diameter (d) of the string:
A = π * (d/2)^2
Where:
d is the diameter of the string (in meters)
Given that the diameter (d) of the guitar string is 0.15 mm, we need to convert it to meters by dividing it by 1000:
d = 0.15 mm / 1000 = 0.00015 m
Now we can find the cross-sectional area (A):
A = π * (0.00015 / 2)^2 = 1.767 x 10^-8 square meters
The Young's modulus of steel is typically around 200 x 10^9 pascals.
Now we have all the values to calculate the spring constant (k):
k = (1.767 x 10^-8) * (200 x 10^9) / 0.73
k = 4.82 x 10^6 N/m
Finally, we can calculate the amount of elongation (x) of the guitar string:
x = F / k
x = 1500 N / (4.82 x 10^6 N/m) = 0.00031 meters or 0.31 mm
Therefore, the steel guitar string stretches by approximately 0.31 mm under the given tension.