# Mass hanging by spring tracing a eight shaped curve

**1. The problem statement, all variables and given/known data**

A mass ##m## on the end of a light spring of force constant ##k## stretches the spring to a length ##l## when at rest. The mass is now set into motion so it executes up and down vibrations while swinging back and forth as a pendulum. The mass moves in a figure-eight pattern in a vertical plane, as shown in the figure. Find the force constant in terms of ##m##,##l## and ##g##.

(Ans: k=4mg/l )

**2. Relevant equations**

**3. The attempt at a solution**

I noticed that the curve traced by the hanging mass is of the form ##r^2=a\cos(2\theta)## with the mass ##m## being at the origin at ##t=0##. But I don’t think this is going to help. This is a question from one of my practice sheets and I doubt I need to deal with polar curves which aren’t generally taught in high school.

How do I approach this problem? :confused:

Any help is appreciated. Thanks!

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