A pilot, whose mass is 91.0 kg, makes a loop-the-loop in a fast jet. Assume that the jet maintains a constant speed of 125 m/s and that the radius of the loop-the-loop is 0.478 km. What is the apparent weight that the pilot feels (i.e., the force with which the pilot presses against the seat) at the bottom of the loop-the-loop?
What is the pilot's apparent weight at the top of the loop-the-loop?
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ApWeight = mg±mv²/R
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Well, well, well, the pilot is taking us for a spin, quite literally! Let's start with the apparent weight at the bottom of the loop-the-loop, shall we?
We need to consider the forces acting on the pilot at that point. At the bottom, the force of gravity is acting downwards as always, but we also have the centrifugal force pushing the pilot towards the center of the loop. The apparent weight will be the sum of these two forces.
So, how do we calculate the centrifugal force, you ask? It's quite simple, my friend. The formula is Fc = m * v^2 / r, where m is the pilot's mass, v is the velocity, and r is the radius of the loop.
Plugging in the numbers, we get Fc = 91.0 kg * (125 m/s)^2 / 0.478 km. Mind the units here! We need to convert that 0.478 km to meters, which gives us 478 meters. Crunching those numbers, we find that the centrifugal force is approximately 3,638 Newtons.
Now, for the bottom of the loop, the pilot's apparent weight will be the sum of the centrifugal force and the force of gravity. Since the force of gravity is always downwards, it remains constant at 91.0 kg * 9.8 m/s^2, which is approximately 892 Newtons.
Adding the two forces together, we get the apparent weight at the bottom of the loop: 3,638 Newtons + 892 Newtons = 4,530 Newtons of pure, gravity-defying weight!
Moving on to the top of the loop-the-loop, things get a bit more interesting. Here the centrifugal force and the force of gravity are acting in the same direction. So, we simply subtract the centrifugal force from the force of gravity.
Using the same equation as before, we find that the centrifugal force is still 3,638 Newtons. However, the force of gravity now becomes 91.0 kg * 9.8 m/s^2, which is approximately 892 Newtons.
Subtracting the centrifugal force from the force of gravity, we find that the apparent weight at the top of the loop-the-loop is a measly 892 Newtons - 3,638 Newtons = -2,746 Newtons! Yes, you heard that right, negative weight! You might even call it weightlessness or anti-gravity!
So, to recap, the pilot feels a whopping 4,530 Newtons of apparent weight at the bottom of the loop-the-loop, and a staggering -2,746 Newtons of apparent weight at the top. These forces are truly out of this world!
To find the apparent weight of the pilot at the bottom and top of the loop-the-loop, we can use the concept of centripetal force.
1. Apparent Weight at the Bottom of the Loop-the-Loop:
At the bottom of the loop, the pilot is moving in a circular path in a vertical plane. The net force acting on the pilot is the difference between their weight and the centripetal force required to keep them in circular motion.
The weight of an object is given by the formula: weight = mass * gravitational acceleration (W = m * g)
At the bottom of the loop, the centripetal force is acting upwards, which is opposite to the direction of gravity. So we subtract the centripetal force from the weight to find the apparent weight:
Apparent weight at the bottom = Weight - Centripetal Force
To find the centripetal force, we use the equation: Centripetal Force = (mass * velocity^2) / radius
Given:
- Mass of the pilot = 91.0 kg
- Velocity of the jet = 125 m/s
- Radius of the loop-the-loop = 0.478 km = 0.478 * 1000 = 478 m
First, let's calculate the centripetal force:
Centripetal Force = (mass * velocity^2) / radius
Centripetal Force = (91.0 * 125^2) / 478
Now, we can calculate the apparent weight at the bottom:
Apparent weight at the bottom = Weight - Centripetal Force
Apparent weight at the bottom = 91.0 * 9.8 - [(91.0 * 125^2) / 478]
2. Apparent Weight at the Top of the Loop-the-Loop:
At the top of the loop, the pilot is moving in a circular path in a vertical plane. The net force acting on the pilot is the sum of their weight and the centripetal force required to keep them in circular motion.
At the top of the loop, the centripetal force is acting downwards, which is in the same direction as gravity. So we add the centripetal force to the weight to find the apparent weight:
Apparent weight at the top = Weight + Centripetal Force
Using the same values as before, calculate the centripetal force and then the apparent weight at the top using the above formula.
Calculating the above equations will give you the apparent weight of the pilot at the bottom and top of the loop-the-loop.