Answer:
a) K = k1k2/k2+k1
b) k1k2k3/(k2k3+k1k3+k1k2)
Explanation:
Hooke's law states that the extension of an elastic material is directly proportional to the applied force provided the elastic limit is not exceeded. Mathematically;
F = ke where;
F is the applied force
k is the elastic constant
e is the extension
If we consider 2 springs 1 an 2 with spring constant k1 and k2 connected in series to each other, their respective spring constant according to hooke's law will be expressed as;
k1 = F/e1 and k2 = F/e2 where F is a constant force.
e1 = F/k1 and e2 = F/k2.
The equivalent extension e = F/K
Since the two spring are in series, the effective spring constant K of the two-spring system is expressed as follows;
Since the total extension of the string
e = e1+e2
F/K = F/k1+F/k2
F(1/K) = F(1/k1)+F(1/k2)
1/K = 1/k1+1/k2
1/K = (k2+k1)/k1k2
Reciprocating both sides gives
K = k1k2/k2+k1
b) Similarly if there are 3 springs connected in series to each other with spring constant k1, k2 and k3, their individual extension will be expressed as;
e1 = F/k1
e2 = F/k2
e3 = F/ke
Their equivalent extension e in series will be expressed as e = F/K
Writing their equivalent extension in terms of their individual extension will give;
e = e1+e2+e3
F/K = F/k1+F/k2+F/k3
1/K = 1/k1+1/k2+1/k3
1/K = (k2k3+k1k3+k1k2)/k1k2k3
Taking the reciprocal of both sides to get K
K = k1k2k3/(k2k3+k1k3+k1k2)