A special prototype model aeroplane of mass 400g has a control wire 8cm long attached to its body. The other end of the control line is attached to a fixed point. When the aeroplane flies with its wings horizontal in a horizontal circle, making one revolution every 4 s, the control wire is elevated 30 degrees above the horizontal. Determine (a)the tension in the control wire, (b)the lift on the plane. (g=10m/s^2 and pi^2=10)
720Newtons
8N
0.08N
To solve this problem, we need to analyze the forces acting on the airplane and use basic principles of circular motion.
Let's start by looking at the forces acting on the airplane when it is flying in a horizontal circle:
1. Weight (mg): This force acts vertically downward. In this case, the weight of the airplane is given by mass (m) multiplied by the acceleration due to gravity (g). So, weight (mg) = 0.4 kg * 10 m/s^2 = 4 N.
2. Tension in the control wire (T): This force acts along the control wire and is directed towards the center of the circular path. We need to determine this tension.
3. Centripetal force (Fc): This force acts radially inward and keeps the airplane moving in a circle. It is provided by the tension in the control wire. We can calculate the centripetal force using the formula Fc = (m * v^2) / r, where v is the velocity and r is the radius of the circular path.
Now, let's calculate the tension in the control wire:
(a) Tension in the control wire:
The plane completes one revolution every 4 s. In this case, the time taken for one revolution (T) is equal to 4 s.
The linear velocity (v) of the airplane is given by v = (2 * π * r) / T, where r is the radius of the circular path.
Since the control wire is elevated 30 degrees above the horizontal, the horizontal component of the velocity is given by v_h = v * cos(30°).
Therefore, v_h = (2 * π * r) / T * cos(30°).
Now, we know that the centripetal force is provided by the tension in the control wire:
Fc = T.
Using the formula for centripetal force, we have:
T = (m * v_h^2) / r.
Substituting the known values:
T = (0.4 kg * ((2 * π * r) / T * cos(30°))^2) / r.
Simplifying the equation:
T = (0.4 kg * 4 * π^2 * cos(30°)^2) / 1.
Since cos(30°) = √3 / 2,
we can simplify the equation further:
T = (0.4 kg * 4 * π^2 * (√3 / 2)^2) / 1.
T = (2.4 kg * π^2 * 3) / 1.
T = 24 π^2 Newtons.
Therefore, the tension in the control wire is approximately 24 π^2 Newtons.
(b) Lift on the plane:
The lift on the plane is equal to the gravitational force because the plane is flying in a horizontal circle.
Therefore, the lift on the plane is 4 N.
To summarize:
(a) The tension in the control wire is approximately 24 π^2 Newtons.
(b) The lift on the plane is 4 Newtons.