A helicopter is using a steel cable to lift a 2100-kg jeep. The unstretched length of the cable is 16 m, and its radius is 5.0 103 m. By what amount does the cable stretch when the jeep is hoisted straight upward with an acceleration of 1.5 m/s2?
To determine the amount by which the cable stretches, we need to consider the tension in the cable and the force exerted on it.
First, we need to calculate the tension in the cable. The tension in the cable is equal to the weight of the jeep plus the force required to accelerate it.
The weight of the jeep is given by the equation:
Weight = mass * acceleration due to gravity
The mass of the jeep is given as 2100 kg and the acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the weight of the jeep is:
Weight = 2100 kg * 9.8 m/s^2
Next, we need to calculate the force required to accelerate the jeep. The force required to accelerate an object is equal to its mass multiplied by the acceleration:
Force = mass * acceleration
In this case, the mass of the jeep is 2100 kg, and the acceleration is 1.5 m/s^2. Therefore, the force required to accelerate the jeep is:
Force = 2100 kg * 1.5 m/s^2
Now, we can calculate the total tension in the cable. Since the cable is being used to lift the jeep, the tension is equal to the weight of the jeep plus the force required to accelerate it:
Tension = Weight + Force
Finally, to determine the amount the cable stretches, we can use Hooke's Law, which states that stress is directly proportional to strain. The stress in the cable is given by:
Stress = tension / cross-sectional area
The cross-sectional area of the cable can be calculated using the formula for the area of a circle:
Area = π * (radius^2)
Now, we can calculate the change in length of the cable using the formula for strain:
Strain = change in length / unstretched length
Since stress is directly proportional to strain, we can rearrange the equation to solve for the change in length:
Change in length = strain * unstretched length
Putting it all together, we can calculate the change in length of the cable.