: The coat hanger shown below is made of PVC (polyvinyl chloride). The neck of the hanger is 0.010 m in diameter and is 0.10 m long. If a coat hangs on the hanger, the length of the neck will increase by 1.00 x10 -5 m.

1. What is the mass of the coat?
2. How many such coats must be hung on the hanger to cause the neck to remain stretched after the coats are removed?

To find the mass of the coat, we can use the information given about the change in length of the PVC neck when the coat hangs on the hanger.

1. What is the mass of the coat?

The change in length of the neck, ΔL, is given as 1.00 x 10^(-5) m. We can use this information to calculate the strain in the neck.

Strain (ε) is defined as the change in length divided by the original length:

ε = ΔL / L₀

where L₀ is the original length of the neck.

We are given the diameter of the neck (d) as 0.010 m. The radius (r) can be calculated by dividing the diameter by 2:

r = d / 2 = 0.010 m / 2 = 0.005 m

The original length of the neck can be calculated using the formula for the circumference of a circle:

L₀ = 2πr

Substituting the value of the radius, we get:

L₀ = 2π(0.005 m) ≈ 0.03142 m

Now we can substitute the given values into the equation for strain:

ε = (1.00 x 10^(-5) m) / (0.03142 m) ≈ 3.18 x 10^(-4)

Next, we can use Hooke's Law to find the stress (σ) in the PVC neck. Hooke's Law states that stress is proportional to strain:

σ = Eε

where E is the elastic modulus of PVC. The value of E for PVC is typically around 3 x 10^9 Pa (pascals).

Substituting the given values into the equation for stress:

σ = (3 x 10^9 Pa)(3.18 x 10^(-4)) ≈ 954 Pa

Finally, we can use the formula for stress to find the force (F) applied to the neck of the hanger when the coat hangs on it. The formula is:

F = Aσ

where A is the cross-sectional area of the neck.

The cross-sectional area of a circle can be calculated using the formula:

A = πr²

Substituting the radius value, we get:

A = π(0.005 m)² ≈ 0.0000785 m²

Finally, substitute the values into the equation for force:

F = (0.0000785 m²)(954 Pa) ≈ 0.0748 N

The force exerted by the coat is approximately 0.0748 Newtons.

Now, let's move on to the second question.

2. How many such coats must be hung on the hanger to cause the neck to remain stretched after the coats are removed?

To cause the neck to remain stretched after the coats are removed, the force exerted by the coats (0.0748 N) must be greater than the elastic restoring force of the neck.

The elastic restoring force can be calculated using Hooke's Law:

F_r = kΔL

where F_r is the elastic restoring force, k is the spring constant (which depends on the material and geometry of the neck), and ΔL is the change in length.

Since we don't know the value of the spring constant, we cannot directly calculate the exact number of coats needed to keep the neck stretched. We would need additional information about the specific material and geometry of the neck.

However, we can make an estimate. Let's assume a hypothetical value for the spring constant (k) as 0.1 N/m. We can now rearrange the equation to solve for the change in length (ΔL):

ΔL = F_r / k

Substituting the values:

ΔL = 0.0748 N / 0.1 N/m = 0.748 m

So, with this hypothetical value for k, approximately 0.748 meters (or 74.8 cm) of change in length would be required to keep the neck stretched.

Therefore, if we assume that each coat causes a change in length of approximately 1.00 x 10^(-5) m (as given in the question), we can estimate the number of coats as follows:

Number of coats = 0.748 m / (1.00 x 10^(-5) m) ≈ 74800 coats

So, approximately 74800 such coats would need to be hung on the hanger to cause the neck to remain stretched after the coats are removed.