In: Physics
Thank you for making the answers clear, so I can rate accordingly, as soon as I get a chance.
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.
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