An 8.0×10^−2 kg toy airplane is tied to the ceiling with a string.?
When the airplane's motor is started, it moves with a constant speed of 1.30 m/s in a horizontal circle of radius 0.40 . Find the angle the string makes with the vertical.
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The string provides two components of force.
#1. Balancing the weight of the aircraft. This is F = m*a =m*g in the vertical direction, or 0.785N
#2. Keeping the aircraft moving in a circular path. F = m*a, where a = (v^2)/r, so F = (m*v^2)/r = 0.338N
These components form the adjacent and opposite sides respectively in a right-angled triangle with the tension in the string. Tan(theta) = opposite/adjacent = 0.338/0/0.785 = 0.43
theta = 23.3 degrees.
Except.. the tension in the string isn't the vector total, or the addition of all the force vectors. If you make a triangle out of that angle on the ceiling, where the side that comes down to the middle of the circle is equal to the weight vector of the plane due to gravity, then make the centripetal force vector the base, and total tension equal to the hypotenuse, the total tension can be found using the Pythagorean theorem. T = sqrt(W^2 + Fcp^2)