Caculating the tention, max accaleration and velocity of a violin string

[mmoadi]

Homework Statement

String on the violin is made of steel wire with a diameter of 0.25 mm. With what force must the string be strained (stretched) so that its basic frequency will equal 660 Hz (E-string)? What is the maximum velocity and acceleration of the parts of the string, if we evoke basic vibration with amplitude of 5 mm? In the rest, the fastened string is long l = 325 mm, the density of steel is 7800 kg/m3.

Homework Equations

μ= m/L
v= 2Lf
a(max)= ω²A

The Attempt at a Solution

First part: With what force must the string be strained so that its basic frequency will equal 660 Hz (E-string)?

Volume of cylinder: V= πr²h= π(1/d)²L= 16 x 10^-9 m³, ρ= 7800 kg/m³
Mass of the string: m= Vρ= 0.00013 kg
Linear mass density: μ= m/L (kg/m)→ μ= 0.0004 kg/m

v= 2Lf= 2(0.325m)(660 hz)= 429 m/s
v= sqrt(T/ μ) → T or F= μv²
F= 73.61 N

Second part: What is the maximum velocity and acceleration of the parts of the string, if we evoke basic vibration with amplitude of 5 mm?

v= 2Lf= 2(0.325m)(660 hz)= 429 m/s
a(max)= ω²A= (2πf)²A= 85897m/s²

 

[chaoseverlasting]

Looks right to me.

 

[mmoadi]

_{Thank you.}