P16_027.PDF

27. We wish to ﬁnd the eﬀective spring constant for the combination of springs shown in Fig. 16–31. We do

this by ﬁnding the magnitude F of the force exerted on the mass when the total elongation of the springs

is ∆ x .Then k eﬀ = F/ ∆ x . Suppose the left-hand spring is elongated by ∆ x and the right-hand spring is

elongated by ∆ x r . The left-hand spring exerts a force of magnitude k ∆ x on the right-hand spring and

the right-hand spring exerts a force of magnitude k ∆ x r on the left-hand spring. By Newton’s third law

these must be equal, so ∆ x =∆ x r . The two elongations must be the same and the total elongation is

twice the elongation of either spring: ∆ x =2∆ x . The left-hand spring exerts a force on the block and

its magnitude is F = k ∆ x .Thus k eﬀ = k ∆ x / 2∆ x r = k/ 2. The block behaves as if it were subject to

the force of a single spring, with spring constant k/ 2. To ﬁnd the frequency of its motion replace k eﬀ in

2 m .

f =

2 π

f =(1 / 2 π ) k eﬀ /m with k/ 2toobtain