Question

1. A powered model airplane of mass 1.77 kg is tied to a ceiling with string and allowed to fly at speed 0.44 m/s in a circular path of radius 0.44 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?

Answer 1

In a uniform circular motion, centripetal force is given by: mv2/r, where m is mass, v is velocity and r is radius.

In the given problem, let the string make an angle with the vertical and let the tension in the string be T.

So,horizontal force on the airplane=Tsin. Now,this force must provide the centripetal force.

So, Tsin=mv2/r....eqn1

Vertical forces on the airplane include

a)gravity=mg downwards, where m is mass and g is gravitational acceleration.

b)force due to tension=Tcos upwards.

Since,there is no vertical acceleration, vertical forces must be balanced=>Tcos=mg....eqn2

Dividing eqn1 by eqn2, Tsin/(Tcos)=(mv2/r)/mg=v2/(rg)

=>tan=v2/(rg). Here,r=0.44 m, v=0.44 m/s.

So,tan=(0.44*0.44)/(0.44*9.8)=0.044898

=>=tan-1(0.044898)=2.57 degrees.

Putting this value of in eqn 2, Tcos2.57=1.77*9.8 (here m=1.77)

=>T=1.77*9.8/(cos2.57)=17.36 N.