A string with a length of 4.00 m is held under a constant tension. The string has a linear mass density of μ = 0.006 kg/m. Two resonant frequencies of the string are 400 Hz and 480 Hz. There are no resonant frequencies between the two frequencies.
(a) What are the wavelengths of the two resonant modes?
(b) What is the tension in the string?

a) λn = 1.6m , λ(n+1) = 1.33

b) F(t) = 2457.6N

Explanation:

L = 4.0m

μ = 0.006kg/m

F1 =F(n) = 400Hz

F2 = F(n+1) = 480Hz

The natural frequencies of normal nodes for waves on a string is

F(n) = n(v / 2L)

F(n) = n / 2L √(F(t) / μ)

where F(t) = tension acting on the string

the speed (v) on a wave is dependent on the tension acting on the string F(t) and mass per unit length μ

v = √(F(t) / μ)

F(n) = frequency of the nth normal node

F(n+1) = frequency of the successive normal node.

frequency of thr nth normal node F(n) = n(v / 2L)equation (i)

frequency of the (n+1)th normal node F(n+1) = (n+1) * (v / 2L)equation (ii)

dividing equation (ii) by (i)

F(n+1) / F(n) = [(n+1) * (v / 2L)] / [n (v / 2L)]

F(n+1) / F(n) = (n + 1) / n

F(n +1) / Fn = 1 + (1/n)

1/n = [F(n+1) / Fn] - 1

1/n = (480 / 400) - 1

1/n = 1.2 - 1

1 / n = 0.2

n = 1 / 0.2

n = 5

the wavelength of the resonant nodes (5&6) nodes are

λ = 2L / n

λn = (2 * 4) / 5

λn = 8 / 5

λn = 1.6

λ(n+1) = (2 * 4) / (5 +1)

λ(n+1) = 8 / 6

λ(n+1) = 1.33m

b.

The tension F(t) acting on the string is

v = √(F(t) / μ)

v² = F(t) / μ

F(t) = μv²

but Fn = n(v / 2L)

nv = 2F(n)L

v = 2F(n)L / n

v = (2 * 400 * 4) / 5

v = 3200 / 5

v = 640m/s

substituting v = 640m/s into F(t) = v²μ

F(t) =(640)² * 0.006

F(t) = 2457.6N

a solenoid is created by wrapping a l = 90 m long wire around a hollow tube of diameter d = 4.5 cm.   the wire diameter is d = 0.9 mm.   the solenoid wire is then connected to a power supply so that a current of i = 9 a flows through the wire.

randomized variablesl = 90 m

d = 4.5 cm

d = 0.9 mm

i = 9 a                                          

write an expression for the number of turns, n, in the solenoid. you do not need to take into account the diameter of the wire in this calculation.

calculate the number of turns, n, in the solenoid.

write an expression for the length of the solenoid (l2) in terms of the diameter of the hollow tube d.

calculate the length of the solenoid (l2)   in meters.

calculate the magnitude of the magnetic field at the center of the solenoid in teslas.

explanation:

1/0.14nm = 10^10/1.4 a lot