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Question

A bead of mass m is attached to one end of a spring of natural length 3R and spring constant k=(3+1)mgR.The other end of the spring is fixed at point A on a smooth fixed vertical ring of radius R as shown in the figure.What is the normal reaction at B just after the bead is released?
1045939_e08a79906b334cb9900cb45e976f034b.png

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Solution

x=2Rcos60=2R12=R
lx=3RR=(3R1)R
Then spring force is
Fs=k(lx)=(3+1)mgR(31)R=2mg
The weight is W=mg
The projection of spring force on the normal is
2mgcos60=2mg12=mg
The projection of weight on the normal is
mgsin60=mg32
The sum of all forces in the normal direction must be zero:
Nmg+mg32=0
N=(132)mg

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