Question

A ball of mass m is tied to a string and is rotating in a vertical plane. The string is elastic (it stretches), which causes the path to be elongated vertically rather than perfectly circular. At the top of the path, the speed has the minimum value that still allows the ball to complete its circular path.

Find: the length of the string when it makes an angle θ with respect to the horizontal.

The following quantities are known:

Mass of the ball , m

Elastic constant of the string, k

Length of the string when the ball is at the top , r0

angle θ

To solve the problem

a) Start by writing the conservation of energy; you know something about the top position, consider that to be your initial; the final state is when the string is at the angle θ. Taking the zero level for potential energy at the center of the circle would make the equation simpler.

b) Once you wrote the equation of conservation of energy, try to express each term as a function of r and known quantities. For example:

- x in the elastic potential energy formula is r-r0;

- The velocity at the top is at a minimum, so you can express it as a function of r0;

For the velocity at the final position, draw a free body diagram, look at the net force along the string and apply Newton’s Second Law along the string. The acceleration is, of course, the centripetal acceleration ;

-h at the final position can be expressed using the angle and r.

Answer 1

If you have any doubt, feel free to ask.

Alright Dude, If that worked for you... dont forget to give THUMBS UP.(that will work for me!)

Thank you!