Question

Figure 14.26 (a) shows a spring of force constant k clamped rigidly at one end anda mass m attached to its free end. A force F applied at the free end stretches thespring. Figure 14.26 (b) shows the same spring with both ends free and attached toa mass m at either end. Each end of the spring in Fig. 14.26(b) is stretched by thesame force F. (a) What is the maximum extension of the spring in the two cases ? (b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case ?

Answer 1

(a)

It is given that force on mass, m is F and force constant of spring is k.

The formula of magnitude of force on spring in figure (a) is,

F=kx

From the above equation, the maximum extension in the spring is,

x= F k

From figure (b), it is clear that both ends of the spring are stretched by the same force as figure (a).

So, the maximum extension in the spring of figure (b) is,

x= F k

Hence, the value of maximum extension produced by spring in both the cases is x= F k .

(b)

The formula of force on the spring is,

F=−kx ma=−kx a=− k m x

Simplify the above equation as,

ω 2 = k m ω= k m

From the formula of time period,

T= 2π ω

Substitute the value of ω in the above equation.

T=2π m k (1)

The spring constant for figure (b) is,

k'=2k

Substitute the value of k' in equation (1).

T=2π m k' T=2π m 2k

Hence, the value of time period for figure (a) is

T=2π m k and for figure (b) is T=2π m 2k