Question

A small 0.501-kg object moves on a frictionless horizontal table in a circular path of radius 1.16 m. The angular speed is 5.51 rad/s. The object is attached to a string of negligible mass that passes through a small hole in the table at the center of the circle. Someone under the table begins to pull the string downward to make the circle smaller. If the string will tolerate a tension of no more than 112 N, what is the radius of the smallest possible circle on which the object can move?

Answer 1

Let r2 is smallest possible radius. and w2 is the anglar speed at this case.

Apply, conservation of angular momentum,

I1*w1 = I2*w2

m*r1^2*w1 = m*r2^2*w2

m*r1^2*v1/r1 = m*r2^2*v2/r2

r1*v1 = r2*v2

r2*v2 = r1^2*w1

= 1.16^2*5.51

= 7.414

T = m*v2^2/r2

= m*v2^2*r2^2/r2^3

= m(v2*r2)^2/r2^3

112 = 0.501*(7.414)^2/r2^3

r2^3 = 0.501*7.414^2/112

r2^3 = 0.2459

r2 = 0.2459^(1/3)

= 0.626 m <<<<<<<<<<-------------Answer