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)
To determine the tension in the control wire and the lift on the plane, we can break down the problem into different components.
(a) Tension in the control wire:
The tension in the control wire can be found using the centripetal force acting on the plane. In a circular motion, the centripetal force is given by the equation:
F = (m * v^2) / r
Where:
F = Centripetal force (tension in the control wire)
m = Mass of the plane (0.4 kg)
v = Velocity of the plane in its circular path
r = Radius of the circular path (length of the control wire)
We are given that the plane makes one revolution every 4 seconds. In one revolution, the plane will travel a distance equal to the circumference of its circular path. The circumference of a circle is given by the formula:
C = 2 * pi * r
Since the plane completes one revolution every 4 seconds, the velocity of the plane is given by:
v = C / t
Substituting the values, we have:
v = (2 * pi * r) / 4
Now, we can substitute this velocity value into the centripetal force equation:
Tension (F) = (m * ((2 * pi * r) / 4)^2) / r
Simplifying further:
F = (m * 4 * pi^2 * r) / 16
Given that pi^2 = 10, we can substitute it, and the mass and radius values:
F = (0.4 * 4 * 10 * r) / 16
Simplifying:
F = (2 * r) / 4
F = r / 2
Therefore, the tension in the control wire is equal to half the radius of the circular path.
(b) Lift on the plane:
The lift on the plane can be found using the equation:
L = m * g * sin(theta)
Where:
L = Lift
m = Mass of the plane (0.4 kg)
g = Acceleration due to gravity (10 m/s^2)
theta = Angle of inclination above the horizontal (30 degrees)
Substituting the values, we have:
L = 0.4 * 10 * sin(30)
Using the value of sin(30) as 1/2, we simplify:
L = 0.4 * 10 * 1/2
L = 2 N
Therefore, the lift on the plane is 2 Newtons.