An elastic cord vibrates with a frequency of 3.5Hz when a mass of 0.62kg is hung from it.What is its frequency if only 0.40kg hangs from it?

To solve this problem, we can use the equation for the frequency of a vibrating string or cord. The frequency is inversely proportional to the square root of the tension, and directly proportional to the square root of the mass per unit length.

First, let's find the frequency with the initial mass of 0.62 kg. We have the frequency (f1) as 3.5 Hz and the mass (m1) as 0.62 kg.

Next, we need to find the frequency with the new mass of 0.40 kg. Let's call this frequency f2, and the mass m2.

Using the equation for the frequency of a vibrating string, we have:

f1 = k√(T/m1)
f2 = k√(T/m2)

where k is a constant and T is the tension in the string.

Since the tension remains the same in both cases, we can set the two expressions equal to each other:

k√(T/m1) = k√(T/m2)

Now, let's solve for f2:

f2 = f1 * √(m1/m2)

Substituting the given values, we get:

f2 = 3.5 Hz * √(0.62 kg / 0.40 kg)

Simplifying further:

f2 = 3.5 Hz * √(1.55)

f2 ≈ 3.5 Hz * 1.244

f2 ≈ 4.34 Hz

Therefore, the frequency of the cord when only 0.40 kg hangs from it is approximately 4.34 Hz.

To determine the new frequency when a different mass is hung from the elastic cord, we can use the equation for the frequency of a vibrating string:

f = (1/2π) √(T/μ)

Where:
f = frequency of vibration
π = pi (approximately 3.14159)
T = tension in the string
μ = mass per unit length of the string

Given that the frequency of vibration (f₁) when a mass of 0.62 kg is hung from the cord is 3.5 Hz, we can calculate the initial tension (T₁) and mass per unit length (μ₁) using the equation:

f₁ = (1/2π) √(T₁/μ₁)

Solving for T₁:

T₁ = (4π²f₁²μ₁)

Now, let's calculate the initial tension (T₁):

T₁ = (4π² * (3.5 Hz)² * μ₁)

Next, we'll calculate the mass per unit length (μ₂) when only 0.40 kg hangs from the cord:

μ₂ = (mass / length)

Given that the mass per unit length of the cord remains constant, we can use the equation μ₁ = μ₂:

μ₂ = μ₁

Now, we can solve for the new frequency (f₂) using the equation f = (1/2π) √(T/μ):

f₂ = (1/2π) √(T₂/μ₁)

Substituting T₂ = T₁ and μ₁ = μ₂:

f₂ = (1/2π) √(T₁/μ₂)

Now we can calculate the new frequency (f₂) by substituting the known values into the equation:

f₂ = (1/2π) √(T₁/μ₂)

f₂ = (1/2π) √(T₁/(0.40 kg))

Finally, we can calculate the new frequency f₂.