In 2014, a thought-provoking question on the qualifying exam for the US Physics Team stirred up discussions among students and educators. The scenario involved a helicopter flying horizontally at a constant speed with a flexible cable hanging beneath it. The challenge was to determine the shape of the cable as the helicopter moved through the air, considering the effects of air friction. The options were:
This article delves into the physics behind this question and the experiments conducted to find the correct answer.
To solve the mystery, a team conducted real-world experiments using a helicopter. They rented a helicopter and used a 20-kilogram, 15-meter-long battle rope. The main concern was to ensure the rope did not interfere with the helicopter’s rotor, which could be dangerous.
As the helicopter flew, the team observed the rope’s behavior. Initially, the rope began to whip due to the rotor wash and surrounding air.
When the team asked viewers on YouTube about the cable’s shape, most chose option C. However, discussions with Professor Paul Stanley, the question’s author, showed that even faculty members had differing opinions. Various homemade experiments led to different conclusions.
During the helicopter flight, the team kept the rope on the right side for safety. As they flew diagonally forward and to the left, it was clear that the rope hung diagonally, confirming option B as the correct answer.
To understand why the rope took this shape, we need to consider the forces involved. Two main forces act on the rope: gravity, pulling it downward, and air resistance, pushing it to the left as the helicopter moves forward. For the rope to maintain a constant speed, these forces must be balanced by the tension in the rope.
The tension varies along the rope’s length, being zero at the bottom and increasing linearly to a maximum at the top. This variation explains why the bottom of the rope wiggles while the top remains steadier. The consistent ratio of air resistance to weight along the rope results in a straight diagonal line when the helicopter flies at a constant speed.
Curious about how adding weight would affect the rope’s shape, the team attached a $20$-pound kettlebell to the end. As the helicopter flew at nearly $100$ kilometers per hour, the rope formed an inverted J shape, corresponding to option D. This shape resembled the original scenario that inspired the exam question, which involved a weighted cable.
The reason for this shape lies in the increased tension needed to support the kettlebell’s weight. As the weight increased, the tension at the bottom of the rope needed to be almost vertical to counteract the weight, while the tension higher up adjusted to balance the increasing air resistance.
To further explore air resistance effects, the team added a Veritasium flag to the rope’s end. Initially, the flag did not significantly change the rope’s shape. However, when a parachute was added, the rope formed a J shape, corresponding to option C. The parachute created substantial air resistance without adding much weight, causing the tension to become nearly horizontal.
This experiment showed that the rope’s shape could vary based on the objects attached to its end. Depending on the weight and air resistance of the load, the rope could take on shapes corresponding to options B, C, or D.
The helicopter experiments provided valuable insights into the physics of a suspended cable. The correct answer to the original question was confirmed as option B when the rope was unweighted. However, adding weight or objects with significant air resistance could lead to different shapes, illustrating the complex interplay of forces at work.
This exploration not only clarified a challenging physics question but also highlighted the importance of hands-on experimentation in understanding fundamental physics concepts.
Gather materials such as a small toy helicopter, string, and weights. Simulate the helicopter’s flight by moving the toy horizontally while observing the string’s behavior. Experiment with different weights and air resistance (e.g., attach a small paper flag) to see how the string’s shape changes. Record your observations and compare them to the scenarios discussed in the article.
Create vector diagrams to represent the forces acting on the cable. Include gravity, air resistance, and tension. Use these diagrams to explain why the cable takes on different shapes under various conditions. Discuss your findings with classmates to deepen your understanding of the forces involved.
Using the principles of physics, derive the equations that describe the forces acting on the cable. Calculate the tension at different points along the cable’s length. Use these calculations to predict the cable’s shape under different conditions, such as varying helicopter speeds or added weights.
Form small groups and discuss the different shapes the cable can take. Debate which factors are most influential in determining the cable’s shape. Consider the role of air resistance, weight, and tension. Present your group’s conclusions to the class and compare them with the article’s findings.
Use computer software to create a simulation of the helicopter and cable scenario. Adjust variables such as speed, weight, and air resistance to observe how they affect the cable’s shape. Share your simulation with the class and explain how it helps visualize the physics concepts discussed in the article.
Helicopter – A type of aircraft that derives both lift and propulsion from one or more sets of horizontally revolving overhead rotors – The helicopter hovered steadily in the air, demonstrating the principles of lift and thrust.
Cable – A strong rope or wire used to bear mechanical loads or transmit electricity – The cable supporting the suspension bridge must withstand significant tension forces.
Air – The invisible gaseous substance surrounding the Earth, a mixture mainly of oxygen and nitrogen – The density of air decreases with altitude, affecting the lift generated by an aircraft’s wings.
Resistance – A force that opposes the motion of an object through a medium – The resistance experienced by a car moving through air is known as aerodynamic drag.
Gravity – The force by which a planet or other body draws objects toward its center – The acceleration due to gravity on Earth is approximately $9.8 , text{m/s}^2$.
Tension – The force exerted by a string, rope, cable, or similar object when it is pulled tight by forces acting from opposite ends – The tension in the cable was calculated using the formula $T = mg + ma$, where $m$ is mass and $a$ is acceleration.
Weight – The force exerted on a body by gravity, calculated as the mass of the body times the local acceleration due to gravity – An astronaut’s weight on the Moon is less than on Earth due to the lower gravitational pull.
Shape – The external form or appearance characteristic of someone or something; the outline of an area or figure – The aerodynamic shape of the car reduces air resistance, improving fuel efficiency.
Forces – Interactions that, when unopposed, will change the motion of an object – The net force acting on an object is the vector sum of all individual forces acting on it, as described by Newton’s second law: $vec{F}_{text{net}} = mvec{a}$.
Experiments – Scientific procedures undertaken to make a discovery, test a hypothesis, or demonstrate a known fact – The physics experiments conducted in the lab helped students understand the principles of electromagnetism.
