A 85-kg snow skier is being pulled up a 17 ° slope by a steel (Young's modulus 2.0 x 1011 N/m2) cable. The cable has a cross-section area of 7.4 x 10-5 m2. The cable applies a force to the skier, and, in doing so, the cable stretches by 2.6 x 10-4 m. A frictional force of magnitude 58 N acts on the skis and is directed opposite to the skier's motion. If the skier's acceleration up the slope has a magnitude of 1.3 m/s2, what is the original (unstretched) length of the cable?

To find the original (unstretched) length of the cable, we need to use Hooke's Law to determine the force exerted by the cable and then use this force to calculate the cable's length.

1. Calculate the force exerted by the cable:
Hooke's Law states that the force exerted by a spring-like object is directly proportional to its displacement from its equilibrium position. In this case, the cable is being stretched by 2.6 x 10^-4 m.

The formula for Hooke's Law is F = k * Δx, where F is the force, k is the spring constant (related to Young's modulus), and Δx is the displacement.

Given:
Young's modulus (steel cable) = 2.0 x 10^11 N/m^2
Cross-sectional area (cable) = 7.4 x 10^-5 m^2
Displacement (cable stretch) = 2.6 x 10^-4 m

The formula for Young's modulus is Y = F/A * L0, where Y is Young's modulus, F is the force applied, A is the cross-sectional area, and L0 is the original length of the cable.

Rearranging the formula to solve for force (F), we get:
F = Y * A * Δx / L0

Substituting the given values:
F = (2.0 x 10^11 N/m^2) * (7.4 x 10^-5 m^2) * (2.6 x 10^-4 m) / L0

2. Calculate the net force on the skier:
The net force acting on the skier is the sum of the force exerted by the cable and the frictional force opposing the skier's motion.

Given:
Frictional force (opposing motion) = 58 N

The formula for the net force is F_net = F_cable - F_friction, where F_net is the net force, F_cable is the force exerted by the cable, and F_friction is the frictional force.

Substituting the calculated force from step 1 and the given frictional force:
F_net = (2.0 x 10^11 N/m^2) * (7.4 x 10^-5 m^2) * (2.6 x 10^-4 m) / L0 - 58 N

3. Calculate the mass of the skier:
The force exerted by the cable is given by F_cable = m * a, where m is the mass of the skier and a is the acceleration.

Rearranging the formula to solve for mass (m):
m = F_cable / a = F_net / a

Substituting the net force and acceleration:
m = (F_net) / (1.3 m/s^2)

4. Calculate the weight of the skier:
Weight (W) = mass * acceleration due to gravity (g)

Given:
Acceleration due to gravity (g) = 9.8 m/s^2

Substituting the calculated mass and acceleration due to gravity:
W = m * g = (F_net) / (1.3 m/s^2) * 9.8 m/s^2

5. Calculate the force exerted by the cable using the weight of the skier:
F_cable = F_net + F_friction = W + F_friction

6. Solve for the original (unstretched) length of the cable:
L0 = F_cable / (Y * A)

Substituting the calculated force and given values:
L0 = (F_cable) / ((2.0 x 10^11 N/m^2) * (7.4 x 10^-5 m^2))

Now, plug in the values calculated in steps 1, 4, and 6 to find the original length of the cable (L0).