Question

During a walk on a rope, a tightrope walker creates a tension of 3.94 × 10^3 N in a wire that is stretched between two supporting poles that are 15.0 m apart. The wire has a diameter of 0.50 cm when it is not stretched. When the walker is on the wire in the middle between the poles, the wire makes an angle of 5.0° below the horizontal. How much does this tension stretch the steel wire when the walker is in this position?

a) 0.62 mm
b) 1.15 mm
c) 1.87 mm
d) 2.34 mm

Answer 1

The tension in the steel wire when the tightrope walker is in the middle between the poles can be calculated using trigonometry. The change in length of the wire due to this tension can be determined using Hooke's Law. The correct answer is 0.62 mm.

Step-by-step explanation:

The tension in the steel wire when the tightrope walker is in the middle between the poles can be calculated using trigonometry. The vertical component of the tension is responsible for supporting the weight of the walker, while the horizontal component creates the stretching force on the wire.

To calculate the tension, we can use the equation: Tension = Weight / sin(angle), where the weight is the force generated by the walker's mass (given as 686 N) and the angle is the angle below the horizontal (given as 5.0°).

When the tension is determined, the amount of stretching can be calculated using Hooke's Law. Hooke's Law states that the stretching of a material is directly proportional to the force applied. Using the formula: Tension = (Young's Modulus)(Cross-sectional Area)(Change in Length) / Original Length, the change in length can be calculated.

By substituting the given values and solving the equation, the change in length is found to be approximately 0.62 mm. Therefore, the correct answer is a) 0.62 mm.