Question

An airplane flies in a loop (a circular path in a vertical plane) of radius 150 m. The pilot's head always points toward the center of the loop. The speed of the airplane is not constant; the airplane goes slowest at the top of the loop and fastest at the bottom. (a) What is the speed of the airplane at the top of the loop, where the pilot feels weightless? (b) What is the apparent weight of the pilot at the bottom of the loop, where the speed of the airplane is 280 km/h? His true weight is 700 N.

Short answer

(a) 38.4 m/s; (b) 3500 N

Step-by-step solution

Understanding Weightlessness at the Top

At the top of the loop, the pilot feels weightless if the only force acting on him is the gravitational force. This occurs when the centripetal force required for circular motion equals the force of gravity acting on the pilot.

Equation of Motion at the Top of the Loop

At the top of the loop, the necessary centripetal force is provided entirely by gravity, hence we have: \[ mg = \frac{mv^2}{r} \]. Simplifying this equation gives us: \[ g = \frac{v^2}{r} \], where \( g = 9.8 \, m/s^2 \) is the acceleration due to gravity and \( r = 150 \, m \) is the radius of the loop.

Solving for Velocity at the Top

Rearrange the equation \( g = \frac{v^2}{r} \) to solve for the speed \( v \): \[ v = \sqrt{gr} = \sqrt{9.8 \times 150} = \sqrt{1470} \]. Calculating this, \( v \approx 38.4 \, m/s \) is the speed at the top of the loop.

Converting Speed Units

First, convert the airplane's speed at the bottom of the loop from km/h to m/s. The given speed is 280 km/h which equals \( \frac{280 \times 1000}{3600} = 77.78 \, m/s \).

Calculating Apparent Weight at the Bottom

At the bottom of the loop, the centripetal force is provided by the normal force (apparent weight) minus the gravitational force. The equation of motion is \( N - mg = \frac{mv^2}{r} \), where \( N \) is the apparent weight. Rearranging gives \( N = mg + \frac{mv^2}{r} \).

Substituting Known Values

Plug in known values for the pilot's weight at the bottom of the loop: \[ N = 700 + \frac{700 \times 77.78^2}{9.8 \times 150} \]. Calculating inside the parentheses: \( \frac{700 \times 77.78^2}{9.8 \times 150} \approx 2800 \). Therefore, \( N = 700 + 2800 = 3500 \, N \).

Key concepts

Centripetal Force

Centripetal force is a key concept in circular motion. It acts on an object moving in a circular path and is directed toward the center of the circle. This isn't a new or separate force; rather, it can be the result of gravitational force, tension, normal force, or a combination of these.

In the context of a pilot flying in a vertical loop, the centripetal force required to keep the pilot moving in a circle is primarily provided by gravity when at the top of the loop, and by normal force at the bottom.

Consider the equation for centripetal force, which is given by \[ F_c = rac{mv^2}{r} \]Where:

• \( F_c \) is the centripetal force,
• \( m \) is the mass,
• \( v \) is the velocity,
• \( r \) is the radius of the circle.

This equation helps us determine the velocity needed at different points in a loop, such as when achieving weightlessness at the top.
Understanding this force helps us analyze the motion in vertical loops and evaluate forces acting at various points.

Apparent Weight

Apparent weight differs from true weight when an object is accelerating, particularly in a non-linear direction like in a circular motion. The apparent weight is the normal force exerted by the seat of the airplane on the pilot, and it changes throughout the loop.

At the bottom of the loop, the apparent weight is greater than the true weight. This increase is due to the pilot experiencing additional forces from the airplane's acceleration, which supports the centripetal force.
The formula \[ N = mg + rac{mv^2}{r} \] illustrates this, showing how apparent weight is influenced by both gravitational force \( mg \) and centripetal force \( \frac{mv^2}{r} \).

• \( N \) is apparent weight.
• As speed and radius change, the apparent weight also adjusts, depending on the loop segment.

Clarifying how apparent weight interacts with the dynamics of a loop helps understand situations including safe speeds and pressure points in flight maneuvers.

Weightlessness

Weightlessness is a sensation encountered in various conditions such as free-fall or specific points in circular movement. In the pilot's loop scenario, weightlessness occurs when only gravitational force supplies the necessary centripetal force.

At the loop's peak, both gravity and the airplane's curvature contribute to this state, aligning the gravitational force to fully sustain the circular trajectory.
When the formula \[ g = rac{v^2}{r} \] is satisfied, weightlessness is felt. Here,

• \( g \) is acceleration due to gravity (about \(9.8 \ m/s^2\)).
• \( v \) represents the speed required.

This relationship implies the speed must be just right such that centripetal needs are entirely managed by gravity. Breaking it down simplifies how individuals onboard might experience zero-g sensations in various trajectories.

Vertical Loop Dynamics

In a vertical loop, dynamics change at each point of motion as different forces come into play. These loops are common in aerobatics, roller coasters, and other thrilling experiences.
Analyzing dynamics involves understanding how the plane's speed changes and how forces are managed at different angles of the loop.

• At the top of the loop, the gravitational force may alone supply the centripetal need.
• At the bottom, both velocity and centripetal forces peak, increasing apparent weight.

Vertical loop dynamics are complex yet follow principles of centripetal forces and gravitation. Key factors such as radius sizes and speed control are pivotal. This strategic management directly impacts comfort and safety, making it essential for design and analysis in motion systems.