A 61-kg snow skier is being pulled up a 14 ° slope by a steel (Young's modulus 2.0 x 1011 N/m2) cable. The cable has a cross-section area of 7.0 x 10-5 m2. The cable applies a force to the skier, and, in doing so, the cable stretches by 2.1 x 10-4 m. A frictional force of magnitude 79 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 1.7 m/s2, what is the original (unstretched) length of the cable?

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To find the original (unstretched) length of the cable, we can use Hooke's Law, which states that the force applied on a spring or an elastic material is directly proportional to the displacement or stretch.

Hooke's Law is given by the equation:

F = k * ΔL

where F is the force applied on the cable, k is the spring constant (also known as the modulus of elasticity), and ΔL is the change in length of the cable.

Here, we are given the following information:
- The skier's mass (m) = 61 kg
- The skier's acceleration (a) = 1.7 m/s^2
- The angle of the slope (θ) = 14°
- The frictional force (F_friction) = 79 N
- The modulus of elasticity (E) = 2.0 x 10^11 N/m^2
- The cross-sectional area of the cable (A) = 7.0 x 10^-5 m^2
- The change in length of the cable (ΔL) = 2.1 x 10^-4 m

First, let's calculate the force acting on the skier due to the slope. We can find this by resolving the weight of the skier parallel and perpendicular to the slope.

Weight parallel to slope (W_parallel) = m * g * sin(θ)
W_parallel = 61 kg * 9.8 m/s^2 * sin(14°)

Weight perpendicular to slope (W_perpendicular) = m * g * cos(θ)
W_perpendicular = 61 kg * 9.8 m/s^2 * cos(14°)

Next, let's calculate the net force acting on the skier:
Net force (F_net) = W_parallel - F_friction

Finally, we can calculate the force applied on the cable using the equation F_net = F:
F = F_net

Now, we can substitute these values into the Hooke's Law equation and solve for the unstretched length of the cable:

F = k * ΔL
F = (E * A / L_0) * ΔL

where L_0 is the original length of the cable.

Rearranging the equation to solve for L_0:

L_0 = (E * A * ΔL) / F

Now, substitute the known values and calculate:

L_0 = (2.0 x 10^11 N/m^2 * 7.0 x 10^-5 m^2 * 2.1 x 10^-4 m) / F

Substitute the value of F and calculate the result to find the original length of the cable.