Two springs with spring constants k1 and k1 are connected to each other in serie

Question

Two springs with spring constants k1 and k1 are connected to each
other in series, end to end. The top of the first spring is attached to the ceiling and the second
string is connected and hanging below the first spring. A mass M is connected to the bottom of
the second spring. The acceleration of gravity is g . Answer the following in terms of quantities
given or a subset thereof.

(a) Derive an expression for the effective spring constant of the spring system?
(b) As the mass hangs vertically and sits motionless what is the distance beyond equilibrium
that the system is stretched?
(c) What is the frequency of the small oscillations of the system?
(d) What is the period of the small oscillations.
(e) If the mass were doubled by what factor would the period of the small oscillations
change?

Explanation / Answer

two springs with spring constants k1 and k2 coneected in series      effective spring constant            K' = K1 K2  / K1 + K2     here  k1 = k2  = k             K '=  k  /2      frequency of smalle oscillation of the system             f  = (1/ 2p ) ( m/k ) ^1/2    here  k = effective spring constant    time period of small oscillation            T = 2p( m/k ) ^1/2               if the  mass  were  doubled          T'  = 2p ( 2m/(k /2 ) ) ^1/2                =  2 T  

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