A powered model airplane of mass 1.7 kg is tied to a ceiling with string and allowed to fly at speed 0.21 m/s in a circular path of radius 0.86 m while suspended by the string which makes a constant angle with respect to the vertical.

(a)
What is the angle, in degrees, that the string takes such that the above is true?
(b)
What is the tension in the string, in N?

4.5

To find the angle that the string makes with the vertical, we need to use the concept of centripetal force.

The centripetal force acting on the airplane is provided by the tension in the string. This force always acts towards the center of the circular path.

(a) To find the angle, we can first calculate the centripetal force acting on the airplane.

The centripetal force, Fc, is given by the equation:
Fc = (mass * velocity^2) / radius

Substituting the given values:
Fc = (1.7 kg * (0.21 m/s)^2) / 0.86 m

Solving this equation gives us the centripetal force acting on the airplane.

Now, let's consider the forces acting on the airplane:

1. Weight (mg): The weight acts vertically downward and is equal to the mass of the airplane multiplied by the acceleration due to gravity (9.8 m/s^2).
2. Tension (T): The tension in the string provides the centripetal force to keep the airplane in a circular path.

Since the string makes an angle with respect to the vertical, the vertical component of the tension (T_y) balances the weight (mg), and the horizontal component of the tension (T_x) provides the centripetal force.

We can relate the tension and the angle using the trigonometric identities:
T_y = T * cos(theta)
T_x = T * sin(theta)

Since T_x provides the centripetal force, it should be equal to Fc.

Let's equate the horizontal component of tension and the centripetal force:

T * sin(theta) = Fc

Now, we can substitute the value of Fc that we calculated earlier and solve for theta.

(b) To find the tension in the string, we can use the equation for T_y:

T_y = T * cos(theta)

Substituting the value of T_y, we can solve for T.

By following these steps, we can find the angle and the tension in the string.