Toy airplane tied to the ceiling with a string?

A 0.075 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 1.01 m/s in a horizontal circle of radius 0.35 m. Find the angle the string makes with the vertical and the tension in the string.



Please show my how to solve this problem. Thanks!|||The summation of vertical forces acting on the airplane is



T(cos A) = mg --- call this Equation 1



where



T = tension in the string

A = angle that string makes with the vertical

m = mass of the airplane = 0.075 kg. (given)

g = acceleration due to gravity = 9.8 m/sec^2 (constant)



The summation of horizontal forces acting on the plane are



T(sin A) = Fc



where



Fc = centrifugal force acting on the toy airplane =mV^2/r

V = speed of the toy airplane = 1.91 m/sec (given)

r = radius of horizontal plane where toy is rotating = 0.35 m



and all the other terms are previously defined.



Therefore, the above becomes



T(sin A) = mV^2/r --- call this Equation 2



NOTE from Equation 1 that



T = mg/cos A and substituting this in Equation 2,



(mg/cos A)(sin A) = mV^2/r



since "m" appears on both sides of the equation, it will simply cancel out and noting that "(sin A/cos A) = tan A" then the above simplifies to



g(tan A) = V^2/r



Substituting values,



(9.8)(tan A) = (1.01)^2/(0.35)



tan A = 1.01^2/(9.8 * 0.35)



tan A = 0.2974



A = arc tan 0.2974



A = 16.56 degrees



Solving for the tension in the string, go back to Equation 1



T(cos A) = mg



and substituting appropriate values,



T(cos 16.56) = 0.35(9.8)



T = 3.58 N



Hope this helps.

|||Pointy is correct for the most part, but when solving for T at the end, when he/she uses T(cos A) = mg, he/she uses .35 (the radius) instead of .075kg (the mass). so it should be T(cos 16.56) = .075(9.8)



T=.767

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|||This is a balance of forces problem.



ID all the forces: gravity and string tension. As the airplane is at a constant altitude, the force of gravity W = Tv the vertical tension in the string. Thus, W = mg = T cos(theta) = Tv; where theta = ? the angle you are looking for. m = .075 kg and g = 9.81 m/sec^2.



As the airplane is at a constant radius of rotation centrifugal force C = Th the horizontal tension in the string. Thus C = mv^2/R = T sin(theta) = Th v = 1.01 mps and R = .35 m.



Solve for m, the common factor in both the vertical and horzontal equations. m = T cos(theta)/g and m = T sin(theta) R/v^2. Set the two equations equal T cos(theta)/g = T sin(theta) R/v^2; then cot(theta) = Rg/v^2 and theta = arccot(Rg/v^2), you can do the math.



Once you have theta, find the tension T = mg/cos(theta). And there you are.

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