A 66-kg water skier is being pulled by a nylon (Young's modulus 3.7 x 109 N/m2) tow rope that is attached to a boat. The unstretched length of the rope is 20 m and its cross-section area is 2.8 x 10-5 m2. As the skier moves, a resistive force (due to the water) of magnitude 180 N acts on her; this force is directed opposite to her motion. What is the change in length of the rope when the skier has an acceleration whose magnitude is 0.62 m/s2?

Lf - Li = F Lo/YA

I'm guessing they want net force
ma + 180
plug and solve

To find the change in length of the rope, we can use Hooke's Law which states that the change in length is directly proportional to the force applied and inversely proportional to the cross-sectional area and Young's modulus of the material.

The formula for Hooke's Law is:

ΔL = (F × L0) / (A × Y)

Where:
ΔL is the change in length of the rope
F is the force applied
L0 is the unstretched length of the rope
A is the cross-sectional area of the rope
Y is the Young's modulus of the material

Given:
F = 180 N
L0 = 20 m
A = 2.8 x 10^-5 m^2
Y = 3.7 x 10^9 N/m^2

Plugging in the values into the formula:

ΔL = (180 N × 20 m) / (2.8 x 10^-5 m^2 × 3.7 x 10^9 N/m^2)

Simplifying:

ΔL = 1.286 x 10^3 m

Therefore, the change in length of the rope when the skier has an acceleration of 0.62 m/s^2 is approximately 1.286 kilometers.

To find the change in length of the rope, you need to use Hooke's Law, which states that the change in length of an object is directly proportional to the applied force and inversely proportional to the material's elasticity.

The formula for Hooke's Law is:

ΔL = (F * L0) / (A * E)

where:
ΔL = change in length of the rope
F = force applied to the rope
L0 = original length of the rope
A = cross-sectional area of the rope
E = Young's modulus of the material

In this case, the force applied to the rope is the resistive force due to water, which is 180 N. The original length of the rope is 20 m. The cross-sectional area is 2.8 x 10^-5 m^2, and the Young's modulus is 3.7 x 10^9 N/m^2.

Substituting these values into the formula:

ΔL = (180 * 20) / (2.8 x 10^-5 * 3.7 x 10^9)

Now, you can calculate the change in length of the rope.