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Question

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A toy plane with a mass of 1.01 kg is tied to a string and made to travel at a speed of 24 m/s in a horizontal circle with a16-m radius. The person holding the string pulls the plane in, increasing the tension in the string, increasing the speed of the plane and decreasing the radius of the plane's orbit. What is the net work done on the plane if the tension in the string increases by a factor of four and the radius decreases to 12 m.

Explanation / Answer

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) = (24)^2/(16)

tan A = (24)^2/(16) /9.8

tan A = 3.673

A = arc tan 3.673

A = 74.76 degrees

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

T(cos A) = mg

and substituting appropriate values,

T(cos 74.76) = 16(9.8)

T = 596.50 N

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