An airplane ✈ makes a turn which has a banking curve of 30° to the
horizontal with a radius 5m, in 10sec, what is the mass of the airplane, if it has a force of 500N. The power the plane travelled with, and what's the speed of the airplane when it made its turn??
Plz show full step help
To solve this problem, we will use the concept of centripetal force, which is the force required to keep an object moving in a curved path.
Step 1: Calculate the centripetal force:
To find the centripetal force, we use the formula:
F = (m * v^2) / r,
where F is the force, m is the mass, v is the velocity, and r is the radius of the curve.
Given:
Banking angle (θ) = 30°
Radius (r) = 5 m
We know that the banking angle is related to the angle between the vertical axis and the inclined plane. In this case, the angle between the inclined plane and horizontal is also 30°.
The first step is to find the horizontal component of the normal force acting on the airplane.
cos θ = adjacent / hypotenuse,
cos 30° = nh / mg,
nh = mg * cos 30°,
nh = mg * √3/2,
nh = √3/2 * mg,
The second step is to find the vertical component of the normal force acting on the airplane.
sin θ = opposite / hypotenuse,
sin 30° = nv / mg,
nv = mg * sin 30°,
nv = mg * 1/2,
nv = 1/2 * mg.
Step 2: Calculate the force of gravity:
The force of gravity acting on the airplane (weight) is given by:
mg = weight.
Step 3: Calculate the centripetal force:
Given:
F = 500 N (force)
From the previous step, we know that the vertical component of the normal force is equal to the weight, so we can substitute it in the equation:
F = nh + nv,
500 N = √3/2 * mg + 1/2 * mg,
500 N = (√3/2 + 1/2) * mg,
500 N = (2 * √3/2) * mg,
500 N = √3 * mg,
mg = 500 N / √3.
Step 4: Calculate the mass of the airplane:
Now, we can find the mass of the airplane by substituting the value of mg into the equation:
mg = 500 N / √3,
m = (500 N / √3) / g.
Step 5: Calculate the power:
Power is defined as the rate at which work is done or energy is transferred per unit of time. In this case, we can calculate the power using the formula:
Power = Work / Time.
Given:
Radius (r) = 5 m
Time (t) = 10 s
Force (F) = 500 N
The work done in moving in a circular path is given by:
Work = force * distance.
The distance traveled in one complete revolution is equal to the circumference of the circle:
Circumference = 2πr.
Work = F * Circumference,
Work = F * 2πr.
Substituting the values, we get:
Work = 500 N * 2π * 5 m.
Now, we can calculate the power using:
Power = Work / Time,
Power = (500 N * 2π * 5 m) / 10 s.
Step 6: Calculate the speed of the airplane:
To calculate the speed, we need to use the formula:
v = ω * r,
where ω is the angular velocity, and r is the radius.
The angular velocity can be found using:
ω = θ / t,
where θ is the angle turned and t is the time taken.
Given:
Banking angle (θ) = 30°,
Time (t) = 10 s,
Substituting the values, we get:
ω = 30° / 10 s.
Finally, substituting the values of ω and r into the formula, we can find the speed:
v = (30° / 10 s) * 5 m.
Now, calculate m, Power, and v using the given values and formulas.