One end of a piano wire is wrapped around a cylindrical tuning peg and the other end is fixed in place. The tuning peg is turned so as to stretch the wire. The piano wire is made from steel (Y = 2.0x1011 N/m2). It has a radius of 0.51 mm and an unstrained length of 0.65 m. The radius of the tuning peg is 1.6 mm. Initially, there is no tension in the wire, but when the tuning peg is turned, tension develops. Find the tension in the wire when the tuning peg is turned through two revolutions.
To find the tension in the wire when the tuning peg is turned through two revolutions, we need to consider the elongation or stretch in the wire.
The formula for the tension in a stretched wire is:
Tension (T) = (Y * A * ΔL) / L
Where:
- T is the tension in the wire
- Y is the Young's modulus of the material (2.0x10^11 N/m^2 for steel)
- A is the cross-sectional area of the wire
- ΔL is the change in length of the wire
- L is the original length of the wire
First, let's calculate the cross-sectional area of the wire using its radius:
A = π * r^2
Given that the radius of the wire is 0.51 mm (0.51 * 10^-3 m), we can calculate the area:
A = π * (0.51 * 10^-3)^2
Next, let's calculate the change in length of the wire. The tuning peg is turned through two revolutions, which means that the length of the wire increases by the circumference of the tuning peg. The formula for the circumference is:
Circumference = 2 * π * r
Given that the radius of the tuning peg is 1.6 mm (1.6 * 10^-3 m), we can calculate the change in length:
ΔL = 2 * π * (1.6 * 10^-3)
Now, we have all the values to calculate the tension:
Tension (T) = (Y * A * ΔL) / L
Substituting the given values:
Tension (T) = (2.0x10^11 N/m^2) * (π * (0.51 * 10^-3)^2) * (2 * π * (1.6 * 10^-3)) / 0.65
Calculating this expression will give us the tension in the wire when the tuning peg is turned through two revolutions.