Question

Figure 14.26 (a) shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure 14.26 (b) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in Fig. 14.26(b) is stretched by the same force F. 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

Hello buddy!!

Well in diagram one side of the spring is fixed and other side is fixed to the load of mass "m"

Suppose the block is displaced a small distance "x" then it experienced a restoring force as :-
F=-kx
Negative sign indicates that the direction of force is opposite to the direction of displacement.
Thus it undergoes SHM

We know what F=ma
therefore
ma=-kx
a=-kx/m

Let k/m=w^2.
hence w=√(k/m)

We know that time period = 2π/w
=2π√(m/k)

Similarly in second case let the two block undergo a force in opposite direction as shown in figure...
therefore the block 1 would undergo displacement "x" in direction of left, and block 2 will undergo displacement "x" in direction of right..
hence total displacement would be "2x"
the restoring force acting on two blocks would be
F=-2kx
since F=ma
therefore
am=-2kx
a=-2kx/m
let, 2k/m=w^2
therefore w=√(2k/m)

we know that time period =2π/w
=2π√(m/2k)
=π√(2m) /√k


Hope this helps!! ^_^