A fighter pilot dives his plane toward the ground at 485 m/s. He pulls out of the dive on a vertical circle. What is the minimum radius of the circle, so that the normal force exerted on the pilot by his seat never exceeds three times his weight?
Well, if the normal force exerted on the pilot's seat never exceeds three times his weight, he should probably skip the circus and join a weightlifting competition instead! But let's get back to the question.
To determine the minimum radius of the circle, we need to consider the forces acting on the pilot. At the top of the circle, the normal force and the pilot's weight add up to provide the necessary centripetal force for circular motion. At the bottom of the circle, however, the normal force must exceed the pilot's weight to prevent them from falling out of their seat.
Now, let's calculate the minimum radius. First, we need to find the acceleration at the bottom of the circle. Using the centripetal acceleration formula, we have:
a = v² / r
where v is the velocity (485 m/s in this case). Let's plug in the numbers:
a = (485 m/s)² / r
Now, we need to find the net force at the bottom of the circle. The net force is the sum of the normal force (N) and the pilot's weight (mg):
N + mg = m * a
Since we're told that the normal force should never exceed three times the pilot's weight, we can write the equation as:
N ≤ 3mg
Plugging in the expression for the net force, we have:
N + mg ≤ 3mg
Now, let's isolate the normal force:
N ≤ 2mg
Finally, we substitute mg with m * a:
N ≤ 2ma
Now, we can substitute for a using the previous equation:
N ≤ 2 * (485 m/s)² / r
Now, let's substitute for N using the weight equation (N = mg):
mg ≤ 2 * (485 m/s)² / r
Finally, let's solve for the minimum radius:
r ≥ 2 * (485 m/s)² / (g * 3)
And don't forget, "r" stands for radius, not "haha"!
Step 1: Understand the problem.
We need to find the minimum radius of the vertical circle that the fighter pilot can fly without exceeding a normal force on the pilot that is greater than three times his weight. The initial velocity of the plane is given as 485 m/s.
Step 2: Identify the relevant principles.
To solve this problem, we need to consider the forces acting on the pilot at different points in the circular motion. At the bottom of the circle, the net force acting on the pilot is equal to the difference between the gravitational force and the normal force. We can use this relationship to find the minimum radius of the circle.
Step 3: Set up the equation.
At the bottom of the circle, the net force acting on the pilot is given by the equation:
Net force = mg - N
where:
m = mass of the pilot
g = acceleration due to gravity
N = normal force
We also know that the normal force should not exceed three times the weight of the pilot, so we can write:
N ≤ 3mg
Step 4: Solve the equation.
The net force at the bottom of the circle can be expressed as the centripetal force:
Net force = mv² / r
where:
m = mass of the pilot
v = velocity of the plane
r = radius of the circle
Equating the two equations, we have:
mv² / r = mg - N
Step 5: Rearrange the equation.
Rearranging the equation, we get:
mv² / r = mg - 3mg
mv² / r = -2mg
Step 6: Solve for the minimum radius.
To find the minimum radius, we need to find the value of r that satisfies the equation:
mv² / r = -2mg
We can cancel out the mass factor:
v² / r = -2g
Rearranging the equation, we get:
r = - v² / (2g)
Step 7: Substitute the values and calculate.
Plugging in the given values, we have:
v = 485 m/s
g = 9.8 m/s² (approximate value)
r = -(485^2) / (2 * 9.8)
r = -235,225 / 19.6
r ≈ -12,000 m
The minimum radius of the circle is approximately 12,000 meters. However, we should note that negative values do not make physical sense in this context. Therefore, the minimum radius of the circle should be the absolute value of the calculated result, which is 12,000 meters.
So, the minimum radius of the circle that the fighter pilot can fly without exceeding a normal force greater than three times his weight is 12,000 meters.
To find the minimum radius of the circle, we need to consider the forces acting on the pilot when he is at the bottom of the vertical circle. At this point, the pilot is experiencing a downward acceleration due to gravity, and a centripetal acceleration towards the center of the circle.
First, let's calculate the acceleration due to gravity:
g = 9.8 m/s^2 (acceleration due to gravity)
Next, let's calculate the centripetal acceleration. We can use the centripetal acceleration formula:
a_c = v^2 / r
where:
a_c is the centripetal acceleration
v is the velocity of the plane
r is the radius of the circle
Given that the velocity of the plane is 485 m/s, and assuming the pilot doesn't change speed during the dive, we can use this velocity as the centripetal acceleration at the bottom of the circle:
a_c = 485 m/s^2
Now, let's determine the net force acting on the pilot at the bottom of the circle. At this point, the net force can be calculated as the difference between the downward force of gravity and the upward force exerted by the seat (normal force):
F_net = F_gravity - F_seat
Considering that the normal force is equal to the pilot's weight when he is not accelerating vertically, we have:
F_seat = m * g
where:
m is the mass of the pilot
g is the acceleration due to gravity
Given that the normal force is not to exceed three times the weight, we can write:
F_seat ≤ 3 * m * g
We also know that the net force is equal to mass times the centripetal acceleration:
F_net = m * a_c
Setting the two equations for net force equal, we get:
m * a_c = F_gravity - F_seat
Substituting the value of F_seat, we have:
m * a_c = m * g - 3 * m * g
Simplifying, we find:
a_c = -2 * g
Since acceleration is always positive, we can ignore the negative sign. Now, let's substitute the known values:
-2 * 9.8 m/s^2 = -19.6 m/s^2 (centripetal acceleration)
Now, let's solve for the radius of the circle using the centripetal acceleration formula:
a_c = v^2 / r
-19.6 m/s^2 = (485 m/s)^2 / r
Rearranging the equation to solve for r:
r = (485 m/s)^2 / (-19.6 m/s^2)
Calculating this value, we find:
r ≈ 12,052 meters
Therefore, the minimum radius of the circle should be approximately 12,052 meters.