An 85,000 kg stunt plane performs a loop-the-loop, flying in a 260-m-diameter vertical circle. At the point where the plane is flying straight down, its speed is 55 m/s and it is speeding up at a rate of 12 m/s per second. What angle does the net force make with the horizontal? Let an angle above horizontal be positive and an angle below horizontal be negative.

Verified step by step guidance

1. Step 1: Identify the forces acting on the stunt plane at the point where it is flying straight down. These forces include the gravitational force (weight), the normal force from the circular motion, and the net force due to acceleration.
2. Step 2: Calculate the gravitational force acting on the plane using the formula: $F=mg$, where $m$ is the mass of the plane (85,000 kg) and $g$ is the acceleration due to gravity (9.8 m/s²).
3. Step 3: Determine the centripetal force required for circular motion using the formula: $F=\frac{m{v}^2}{r}$, where $v$ is the speed of the plane (55 m/s) and $r$ is the radius of the circle (half the diameter, 130 m).
4. Step 4: Calculate the net force acting on the plane using the formula: $F=ma$, where $a$ is the acceleration (12 m/s²). Combine this with the forces calculated in Steps 2 and 3 to find the total net force vector.
5. Step 5: Use trigonometry to find the angle of the net force with respect to the horizontal. The vertical component of the net force is the sum of the gravitational force and the centripetal force, while the horizontal component is due to the acceleration. Use the formula: $\theta =\tan (\frac{F}{}$_verticalF_horizontal) to calculate the angle, ensuring the sign convention for above or below the horizontal is applied.

Key Concepts

Net Force

Net force is the vector sum of all forces acting on an object. In the context of the stunt plane, it includes gravitational force and the aerodynamic forces acting on the plane. Understanding net force is crucial for determining the resultant direction and magnitude of the forces, which directly affects the plane's motion during the loop.

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path, necessary for an object to maintain circular motion. For the stunt plane, this acceleration is influenced by its speed and the radius of the loop. It plays a key role in calculating the net force acting on the plane at different points in the loop, particularly when it is at the bottom of the loop.

Angle of Net Force

The angle of the net force with respect to the horizontal indicates the direction of the resultant force acting on the plane. This angle can be determined using trigonometric relationships based on the components of the net force, which include both vertical and horizontal forces. Understanding this angle is essential for analyzing the plane's dynamics and ensuring it performs the loop safely.