Pilots of high-performance fighter planes can be subjected to large centripetal accelerations during high-speed turns. Because of these accelerations, the pilots are subjected to forces that can be much greater than their body weight, leading to an accumulation of blood in the abdomen and legs. As a result, the brain becomes starved for blood, and the pilot can lose consciousness (''black out''). The pilots wear ''anti-G suits'' to help keep the blood from draining out of the brain. To appreciate the forces that a fighter pilot must endure, consider the magnitude FN of the normal force that the pilot's seat exerts on him at the bottom of a dive. The magnitude of the pilot's weight is W. The plane is traveling at 232 m/s on a vertical circle of radius 652 m. Determine the ratio FN/W. For comparison, note that blackout can occur for values of FN/W as small as 2 if the pilot is not wearing an anti-G suit.
To solve this problem, we need to analyze the forces acting on the pilot at the bottom of the dive.
The forces involved are the pilot's weight (W) and the normal force (FN) exerted by the seat. The normal force is directed upwards and can be broken down into two components: one providing the necessary centripetal force to keep the pilot moving in a circular path, and the other balancing the weight of the pilot.
Let's start by finding the gravitational force acting on the pilot (W). The weight of an object is given by the formula:
W = m * g
where m is the mass of the pilot and g is the acceleration due to gravity. However, the mass cancels out when calculating the ratio FN/W, so we don't need to know the actual mass of the pilot.
Now, let's find the centripetal force required to keep the pilot moving in a circular path. The centripetal force is given by:
Fc = m * ac
where m is the mass of the pilot and ac is the centripetal acceleration. In this case, the centripetal acceleration can be calculated using the formula:
ac = (v^2) / r
where v is the velocity of the plane and r is the radius of the circular path.
Once we have the centripetal force, we can determine the ratio FN/W by dividing the magnitude of the normal force (FN) by the magnitude of the weight (W).
The equation becomes:
FN / W = (Fc / W) + 1
Let's plug in the given values:
v = 232 m/s (velocity)
r = 652 m (radius)
First, let's find the centripetal force:
ac = (v^2) / r
ac = (232^2) / 652
ac ≈ 82.684 m/s^2
Next, we find the ratio FN/W:
FN / W = (Fc / W) + 1
FN / W = (m * ac) / (m * g) + 1
FN / W = ac / g + 1
FN / W = (82.684 m/s^2) / (9.8 m/s^2) + 1
FN / W ≈ 8.425 + 1
FN / W ≈ 9.425
The ratio FN/W is approximately 9.425.
Therefore, the ratio FN/W at the bottom of the dive is approximately 9.425. This is significant because a ratio as small as 2 can cause a blackout if the pilot is not wearing an anti-G suit.
To determine the ratio FN/W, we need to analyze the forces acting on the pilot when they are at the bottom of the dive.
First, let's consider the forces acting on the pilot at the bottom of the vertical circle:
1. Weight (W): This force acts vertically downward and is equal to the mass of the pilot multiplied by the acceleration due to gravity (W = mg). It represents the gravitational pull on the pilot.
2. Normal Force (FN): This is the force exerted by the pilot's seat on the pilot. At the bottom of the dive, the seat is pushing the pilot upward to counteract their weight. It acts perpendicular to the seat and provides the necessary centripetal force to keep the pilot moving in a circular path.
To find the magnitude of the normal force, we can use Newton's second law, which states that the net force on an object is equal to its mass multiplied by its acceleration. In this case, the net force is the difference between the normal force and the weight.
At the bottom of the dive, the net force acting on the pilot in the vertical direction is given by:
Net Force = FN - W
Since the pilot is moving in a circular path, the net force is equal to the centripetal force required to keep the pilot moving in that path.
Centripetal Force = (mass of pilot) * (centripetal acceleration)
The centripetal acceleration is given by:
Centripetal Acceleration = (velocity of the plane)^2 / (radius of the circle)
Substituting the values given in the problem, we get:
Centripetal Acceleration = (232 m/s)^2 / (652 m) = 82.47 m/s^2
Hence, the centripetal force required is:
Centripetal Force = (mass of pilot) * (82.47 m/s^2)
Now we can equate the net force to the centripetal force:
FN - W = (mass of pilot) * (82.47 m/s^2)
Since W = (mass of pilot) * (acceleration due to gravity), we can substitute it in the equation:
FN - (mass of pilot) * (acceleration due to gravity) = (mass of pilot) * (82.47 m/s^2)
Now, let's rearrange the equation to solve for FN:
FN = (mass of pilot) * (82.47 m/s^2) + (mass of pilot) * (acceleration due to gravity)
Since we are looking for the ratio FN/W, we divide both sides of the equation by W:
FN/W = [(mass of pilot) * (82.47 m/s^2) + (mass of pilot) * (acceleration due to gravity)] / (mass of pilot) * (acceleration due to gravity)
The mass of the pilot cancels out:
FN/W = 82.47 m/s^2 / (acceleration due to gravity)
Now, we can substitute the value of acceleration due to gravity, which is approximately 9.8 m/s^2:
FN/W = 82.47 m/s^2 / 9.8 m/s^2
Finally, we can calculate the ratio:
FN/W ≈ 8.42
Therefore, the ratio FN/W is approximately 8.42.
This means that at the bottom of the dive, the normal force exerted by the seat on the pilot is about 8.42 times the weight of the pilot. This high ratio helps to prevent the pilot from losing consciousness ("black out") by keeping the blood from draining out of the brain.