Problem 12 - College Physics

Question

A \(\mathrm{A} 50.0 \mathrm{kg}\) stunt pilot who has been diving her airplane vertically pulls out of the dive by changing her course to a circle in a vertical plane. (a) If the plane's speed at the lowest point of the circle is \(95.0 \mathrm{m} / \mathrm{s},\) what should the minimum radius of the circle be in order for the centripetal acceleration at this point not to exceed 4.00 \(\mathrm{g}\) ? (b) What is the apparent weight of the pilot at the lowest point of the pullout?

Step-by-Step Solution

Answer

(a) 230.3 m, (b) 2452.5 N.

Step 1: Understanding the parameters

We need to find the minimum radius of the circle using the speed of the airplane at the lowest point of the circle and the maximum centripetal acceleration. The pilot's mass and the acceleration due to gravity are also given: \[ v = 95.0 \, \text{m/s} \] \[ a_c = 4.00 \, g \text{, where} \, g = 9.81 \, \text{m/s}^2 \] \[ m = 50.0 \, \text{kg} \]

Step 2: Calculating maximum centripetal acceleration

Convert the given multiple of gravity to meters per second squared to find the maximum centripetal acceleration:\[ a_c = 4.00 \times 9.81 \frac{\text{m}}{\text{s}^2} = 39.24 \frac{\text{m}}{\text{s}^2} \]

Step 3: Using centripetal acceleration formula to find radius

Use the formula for centripetal acceleration \( a_c = \frac{v^2}{r} \) to solve for the radius \( r \):\[ 39.24 = \frac{95.0^2}{r} \]Solve for \( r \) by rearranging the equation:\[ r = \frac{95.0^2}{39.24} \approx 230.3 \, \text{m} \]

Step 4: Calculating apparent weight at lowest point

At the lowest point of the pullout, the apparent weight (normal force) is the sum of gravitational force and centripetal force:\[ W_\text{apparent} = m(g + a_c) \]\[ W_\text{apparent} = 50.0(9.81 + 39.24) \]\[ W_\text{apparent} = 50.0 \times 49.05 = 2452.5 \, \text{N} \]

Key Concepts

Centripetal Acceleration

Centripetal acceleration is a crucial concept when analyzing objects in circular motion. It occurs because an object changing direction is accelerating, even if its speed remains constant. For an object moving in a circle of radius \( r \) at a speed \( v \), the centripetal acceleration \( a_c \) can be defined by the formula \( a_c = \frac{v^2}{r} \). This formula shows that centripetal acceleration depends on the square of the speed and is inversely proportional to the radius of the circle.

In the context of the stunt pilot scenario, the centripetal acceleration must not exceed a multiple of the gravitational acceleration. Here, it should not exceed \( 4g \), where \( g = 9.81 \, \text{m/s}^2 \). By converting this, we find \( a_c = 39.24 \text{ m/s}^2 \).

Understanding the factors affecting centripetal acceleration helps in designing safe and controlled circular movements, which is particularly of interest in aviation and amusement park ride designs.

Apparent Weight

Apparent weight is the weight perceived by an object due to forces acting on it, distinct from its actual weight. When a pilot pulls out of a dive in a vertical circle, forces combining gravity and centripetal acceleration alter the pilot's apparent weight. At the circle's lowest point, these forces reach a maximum, influencing how heavy the pilot feels.

The formula for apparent weight at the lowest point is \( W_\text{apparent} = m(g + a_c) \), where \( m \) is the mass of the pilot. The apparent weight becomes the sum of gravitational and centripetal force, giving a more significant force than simply standing still on the ground: \( 2452.5 \, \text{N} \) in this case.

• This increase makes the pilot feel heavier, a phenomenon experienced as g-forces by pilots and astronauts.
• It's crucial for understanding limits within human tolerance in high-speed maneuvers.

In practical terms, understanding apparent weight is vital for safe pilot training and vehicle design, ensuring maneuvers stay within safe limits.

Circular Motion

Circular motion refers to any motion in which an object moves along a circular path. In physics, this movement is significant due to the constant change in direction, resulting in continuous acceleration and thus a net force toward the center of the circle, known as centripetal force.

To maintain circular motion, a force must continually act at right angles to the object's velocity. In our example, the stunt pilot guides the airplane in a circular path during the pull-out maneuver, applying enough force to balance gravity and achieve the desired path without exceeding safe limits of centripetal acceleration.

Key factors in circular motion:

• The speed of the object must be high enough to generate the needed centripetal force but not so high as to overcome the structural limits or cause harm.
• The radius of the path determines how tight or wide the circle is, affecting the required centripetal force.

Mastering circular motion concepts is essential for scenarios like flying, where maneuverability and control are paramount for both safety and efficiency.