An airplane moves 243 m/s as it travels
around a vertical circular loop which has a
radius of 1.2 km.
What is the magnitude of the normal force
on the 100 kg pilot of this plane at the bottom
of this loop? The acceleration of gravity is
9.8 m/s
2
.
Answer in units of k
M g + M V^2/R
R = 1200 m
"Units of k" makes no sense. The answer will not be a temperature; it will be a force, in Newtons.
It will not be answered in units of k. It is asking for a force, so the units will be in Newtons. (I assume you meant "kN", which would be kiloNewtons.)
The forces acting on the pilot at the bottom of the loop are gravity and the normal force. However, they are pulling in opposite directions (gravity pulls away from the center of the loop, while the normal force pulls toward the center), so the magnitudes of both forces need to be subtracted to get the total net force (F = N - mg).
m = 100 kg
v = 243 m/s
r = 1.2 km (1200 m)
g = 9.8 m/s
With the information given, the total net force can be calculated using the centripetal force formula: F = m * (v^2 / r)
F = 100 * (243^2 / 1200) = 4920.75 N
The problem is only asking for the magnitude of the normal force. As I stated earlier, two forces are acting on the pilot at the bottom of the loop (gravity and the normal force). The force of gravity is calculated with this formula: F = mg
F = 100 * 9.8 = 980 N
Now, take another look at the net force equation.
F = Normal force - mg
Since gravity is pulling away from the center of the loop, it is the negative force, which is why it is subtracted from the normal force to get the net force. So to find the normal force, the force of gravity needs to be added to the net force.
F + mg = Normal force
4920.75 + 980 = 5900.75 N
And since the problem asks for the force in kiloNewtons...
5900.75 N / 1000 = 5.90075 kN
To find the magnitude of the normal force on the pilot at the bottom of the loop, we need to consider the forces acting on the pilot.
At the bottom of the loop, the pilot experiences two forces – gravity (mg) acting downward and the normal force (N) acting upward.
The centripetal force required to keep the plane moving in a circular path at the bottom of the loop is provided by the net force acting on the pilot, which is the vector sum of the normal force and gravitational force.
The net force is given by:
Net Force = Centripetal Force = m * v^2 / r
where m is the mass of the pilot, v is the velocity of the plane, and r is the radius of the loop.
Plugging in the values:
m = 100 kg
v = 243 m/s
r = 1.2 km = 1200 m
Net Force = (100 kg) * (243 m/s)^2 / (1200 m)
Now we can calculate the normal force at the bottom of the loop by subtracting the gravitational force from the net force:
Normal Force = Net Force - gravity
Normal Force = (100 kg) * (243 m/s)^2 / (1200 m) - (100 kg) * (9.8 m/s^2)
Normal Force = (100) * (243^2) / (1200) - (100) * (9.8)
Normal Force = 59.783 kN - 980 N
Normal Force = 59.783 kN - 0.980 kN
Normal Force = 58.803 kN
Therefore, the magnitude of the normal force on the pilot at the bottom of the loop is approximately 58.803 kN.
To find the magnitude of the normal force on the pilot at the bottom of the loop, we'll use the following steps:
1. Calculate the centripetal acceleration: The centripetal acceleration is given by the formula a = v^2 / r, where v is the velocity and r is the radius. Convert the radius from kilometers to meters. So, a = (243 m/s)^2 / (1.2 km * 1000 m/km).
2. Calculate the net force: The net force is the sum of all the forces acting on the pilot. At the bottom of the loop, the net force is the difference between the gravitational force and the centripetal force. The gravitational force is given by F = m * g, where m is the mass of the pilot and g is the acceleration due to gravity.
3. Calculate the normal force: The normal force is the force exerted by the surface on the pilot to counteract the force of gravity. At the bottom of the loop, the normal force is equal to the net force.
Let's plug in the given values and calculate the normal force:
1. Convert radius to meters: 1.2 km * 1000 m/km = 1200 m.
Calculate centripetal acceleration: a = (243 m/s)^2 / 1200 m.
2. Calculate gravitational force: F = 100 kg * 9.8 m/s^2.
Calculate net force: net force = F - m * a.
3. Calculate the normal force: normal force = net force.
After performing these calculations, you will have the magnitude of the normal force in units of kilonewtons (kN).