A copper (Young's modulus 1.1 x 1011 N/m2) cylinder and a brass (Young's modulus 9.0 x 1010 N/m2) cylinder are stacked end to end, as in the drawing. Each cylinder has a radius of 0.14 cm. A compressive force of F = 4800 N is applied to the right end of the brass cylinder. Find the amount by which the length of the stack decreases.

To find the amount by which the length of the stack decreases, we can use Hooke's law, which states that the deformation of an object is directly proportional to the applied force.

First, let's calculate the original lengths of the copper and brass cylinders. We can use the formula for the length of a cylinder:

L = 2 * π * r,

where L is the length and r is the radius.

For the copper cylinder:
L_copper = 2 * π * 0.14 cm = 2 * 3.14 * 0.14 cm = 0.88 cm.

For the brass cylinder:
L_brass = 2 * π * 0.14 cm = 2 * 3.14 * 0.14 cm = 0.88 cm.

Since the copper and brass cylinders are stacked end to end, the initial length of the stack is the sum of their lengths:

L_initial = L_copper + L_brass = 0.88 cm + 0.88 cm = 1.76 cm.

To find the amount by which the length of the stack decreases, we need to determine the stress in the cylinders caused by the applied force.

The stress (σ) in a cylinder is given by:

σ = F / A,

where F is the applied force and A is the cross-sectional area of the cylinder.

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

A = π * r^2,

where r is the radius.

For the copper cylinder:
A_copper = π * (0.14 cm)^2 = 0.0616 cm^2.

For the brass cylinder:
A_brass = π * (0.14 cm)^2 = 0.0616 cm^2.

Now, we can calculate the stress in the cylinders:

σ_copper = F / A_copper = 4800 N / 0.0616 cm^2 = 7798.7 N/cm^2.

σ_brass = F / A_brass = 4800 N / 0.0616 cm^2 = 7798.7 N/cm^2.

Since the stresses are the same for both cylinders (as they are stacked together), the strains in the cylinders will be different, as they depend on the Young's modulus, which is not the same for copper and brass.

The strain (ε) in a material is defined as the ratio of the change in length (ΔL) to the original length (L_initial):

ε = ΔL / L_initial.

We can rearrange this formula to find the change in length:

ΔL = ε * L_initial.

Now, let's calculate the strain for each cylinder:

ε_copper = σ_copper / Y_copper = 7798.7 N/cm^2 / (1.1 x 10^11 N/m^2) = 7.09 x 10^-8.

ε_brass = σ_brass / Y_brass = 7798.7 N/cm^2 / (9.0 x 10^10 N/m^2) = 8.67 x 10^-8.

Finally, we can find the change in length:

ΔL_copper = ε_copper * L_initial = (7.09 x 10^-8) * 1.76 cm = 1.25 x 10^-7 cm.

ΔL_brass = ε_brass * L_initial = (8.67 x 10^-8) * 1.76 cm = 1.53 x 10^-7 cm.

The total change in length of the stack is the sum of the individual changes in length:

ΔL_total = ΔL_copper + ΔL_brass = 1.25 x 10^-7 cm + 1.53 x 10^-7 cm = 2.78 x 10^-7 cm.

Therefore, the amount by which the length of the stack decreases is 2.78 x 10^-7 cm.

To find the amount by which the length of the stack decreases, we need to calculate the total compression of the cylinders. We can find the change in length of each cylinder separately and then add them together.

1. Calculate the initial length of each cylinder:
The initial length of each cylinder can be calculated as follows:
- Length of the copper cylinder (L_copper) = 2πr_copper, where r_copper is the radius of the copper cylinder.
- Length of the brass cylinder (L_brass) = 2πr_brass, where r_brass is the radius of the brass cylinder.

Given that the radius of both cylinders is 0.14 cm, we convert it to meters:
r_copper = r_brass = 0.14 cm = 0.14/100 m = 0.0014 m.

Calculating the initial length:
L_copper = 2π(0.0014 m) = 0.00879 m.
L_brass = 2π(0.0014 m) = 0.00879 m.

2. Calculate the change in length for each cylinder:
The change in length (ΔL) of a cylinder can be calculated using Hooke's Law:
ΔL = (F * L) / (A * E), where F is the applied force, L is the initial length, A is the cross-sectional area, and E is the Young's modulus.

For the copper cylinder:
- Cross-sectional area (A_copper) = πr_copper^2.
- Young's modulus (E_copper) = 1.1 x 10^11 N/m^2.
ΔL_copper = (F * L_copper) / (A_copper * E_copper) = (4800 N * 0.00879 m) / (π(0.0014 m)^2 * (1.1 x 10^11 N/m^2).

For the brass cylinder:
- Cross-sectional area (A_brass) = πr_brass^2.
- Young's modulus (E_brass) = 9.0 x 10^10 N/m^2.
ΔL_brass = (F * L_brass) / (A_brass * E_brass) = (4800 N * 0.00879 m) / (π(0.0014 m)^2 * (9.0 x 10^10 N/m^2).

3. Calculate the total change in length:
To find the total change in length, we simply add the individual changes in length of the two cylinders together:
ΔL_total = ΔL_copper + ΔL_brass.

Now you can substitute the values for the variables and calculate the total change in length.