A 78 cm long, 0.8 mm diameter steel guitar string must be tightened to a tension of 1600 N by turning the tuning screws. By how much is the string stretched?
To find out how much the string is stretched, we need to calculate the change in length.
The formula to calculate the change in length of a string due to tension is:
ΔL = (F * L) / (A * E)
Where:
ΔL is the change in length
F is the tension force
L is the original length of the string
A is the cross-sectional area of the string
E is the Young's modulus of the material
First, let's calculate the original length of the string. The problem states that the string is 78 cm long.
L = 78 cm
Next, we need to calculate the cross-sectional area of the string. The problem states that the diameter of the string is 0.8 mm. We can use the formula for the area of a circle to find the cross-sectional area.
A = π * (d/2)^2
Where:
A is the cross-sectional area
π is a mathematical constant approximately equal to 3.14159
d is the diameter of the string
Converting the diameter to meters:
d = 0.8 mm = 0.8 * 10^-3 m
Now we can calculate the cross-sectional area:
A = π * (0.8 * 10^-3 / 2)^2
Next, we need to determine the Young's modulus of steel. The Young's modulus is a measure of the stiffness of a material. For steel, it's typically around 200 GPa (Gigapascals), which is equivalent to 200 * 10^9 N/m^2.
E = 200 * 10^9 N/m^2
Finally, we can plug the values into the formula to calculate the change in length:
ΔL = (F * L) / (A * E)
= (1600 N * 78 cm) / (π * (0.8 * 10^-3 / 2)^2 * 200 * 10^9 N/m^2)
Now, calculate the value of ΔL using the given values and solve the equation.
To calculate the amount by which the guitar string is stretched, we need to determine its original length and final length under tension, and then subtract the original length from the final length.
1. Convert the diameter from millimeters to meters:
0.8 mm = 0.8/1000 = 0.0008 m
2. Calculate the original length "L" of the steel guitar string:
L = 78 cm = 78/100 = 0.78 m
3. Calculate the area "A" of the steel guitar string:
A = π r^2
= π (d/2)^2
= π (0.0008/2)^2
= π (0.0004)^2
= π (0.00016)
≈ 0.000502 m^2 (rounding to six decimal places)
4. Calculate the tension "T" in the steel guitar string:
T = 1600 N
5. Use Hooke's Law to calculate the final length "Lf" of the steel guitar string under tension:
T = YA (Lf - L)
1600 = Y * 0.000502 * (Lf - 0.78)
Lf - 0.78 = 1600 / (Y * 0.000502)
Lf ≈ 0.78 + (1600 / (Y * 0.000502))
Note: Y is the Young's modulus, which is a measure of the stiffness of the material. For steel, it is typically around 200 GPa (gigapascals).
6. Calculate the stretch of the string by subtracting the original length "L" from the final length "Lf":
Stretch = Lf - L
≈ (0.78 + (1600 / (Y * 0.000502))) - 0.78
Please provide the value of Y (Young's modulus for steel) to get the final numerical answer.