A box is suspended by a steel wire via a frictionless pulley

The wire has a mass per length of µ = 24.5 × 10 − 3 kg/m.
The box is so heavy and the wire so thin that the mass of the wire does not influence the wire tension.
The horizontal part of the wire has a length l = 1 m and it is observed that the lowest frequency for standing waves on this part is f1 = 29.6 Hz.

What is the mass M of the box?

Did you get an answer for this one? Could need one too.

To find the mass of the box, we can start by analyzing the properties of the wire.

The wave equation for a wire under tension is given by:

v = √(T / µ)

where v is the wave velocity, T is the tension in the wire, and µ is the mass per length of the wire.

In this case, we are given the lowest frequency for standing waves on the horizontal part of the wire. The frequency of a standing wave is related to the wave velocity and the length of the wire by the equation:

f = v / λ

where f is the frequency and λ is the wavelength.

Since we know the frequency (f1 = 29.6 Hz) and the length (l = 1 m) of the wire, we can rearrange the equation to solve for the wave velocity:

v = f1 * λ

Now, we need to determine the wavelength of the standing wave on the wire. In this case, the wire has a horizontal part, and we are looking for the fundamental frequency (lowest frequency) standing wave. The fundamental frequency for a standing wave on a wire with both ends fixed is given by:

f1 = (v / 2l)

Rearranging the equation, we can solve for the wavelength:

λ = 2l / f1

Now, substitute the values of l and f1 into the equation to find the wavelength.

Once we have the wavelength, we can substitute it back in the equation v = f1 * λ to find the wave velocity.

Next, we need to calculate the tension in the wire. The tension in the wire is determined by the weight of the box and the force required to support it. In this case, the box is suspended by the wire, so the tension in the wire is equal to the weight of the box:

T = Mg

where T is the tension, M is the mass of the box, and g is the acceleration due to gravity.

Finally, substitute M and g into the equation to solve for the mass M of the box.

Let's go through the calculations step by step:

1. Calculate the wavelength (λ) of the standing wave on the wire:
λ = 2l / f1

2. Calculate the wave velocity (v) of the wire:
v = f1 * λ

3. Calculate the tension (T) in the wire:
T = µ * l * g

4. Solve for the mass (M) of the box:
M = T / g

Substitute the values of µ, l, g, and the calculated values of λ and v into the equations to find the mass M of the box.