# a 10m long steel cable is lifting a 30 ton crate upward off a ship. If the cable streches by .5cm under these conditions, determine the radius of the steel cable. Young's modulus of steel :2x10raised to 9 Pa

## assume metric ton of 1000 kg

tension = m g = 30*10^3 *9.81 Newtons

area = pi r^2

5*10^-3/10 = 30*10^3*9.81 /[pi r^2 *2*10^9]

## To determine the radius of the steel cable, we can use Hooke's Law and the formula for stress:

Stress = (Force / Area)

Where:
- Force is the force applied on the cable (weight of the crate),
- Area is the cross-sectional area of the cable, and
- Stress is the force per unit area.

Given:
- Length of the steel cable (L) = 10 m
- Stretch of the cable (ΔL) = 0.5 cm = 0.005 m
- Weight of the crate (Force) = 30 tons = 30000 kg
- Young's modulus of steel (Y) = 2 × 10^9 Pa

First, we need to calculate the force applied on the cable:

Force = mass × gravity
Force = 30000 kg × 9.8 m/s^2
Force = 294000 N

Next, we can calculate the stress on the cable:

Stress = Force / Area

However, we do not have the value for the cross-sectional area of the cable. To find it, we can use the formula for elongation due to stress:

ΔL = (Stress × L) / (Y × Area)

Rearranging the formula gives us:

Area = (Stress × L) / (Y × ΔL)

Substituting the given values:

Area = (294000 N × 10 m) / (2 × 10^9 Pa × 0.005 m)
Area = 2940000 m² / 10000 m²
Area = 294 m²

Now, we can calculate the radius of the steel cable by dividing the area by π:

Area = π × r²

294 = 3.14159 × r²

Dividing both sides by π:

r² = 294 / 3.14159
r² ≈ 93.38

Taking the square root of both sides:

r ≈ √93.38
r ≈ 9.66 meters

Therefore, the radius of the steel cable is approximately 9.66 meters.