Two rectangular blocks A and B of masses 2 kg and 3 kg respectively are connected by a spring of constant 10.8 Nm^-1 and are placed on a frictionless horizontal surface. The block A was given an initial velocity of 0.15 ms^-1 in the direction shown in the figure. The maximum compression of the spring during the motion is

        

• 0.01 m

• 0.02 m

• 0.05 m

• 0.03 m

C.

0.05 m

As the block A moves with velocity 0.15 ms^-1  it compresses the spring which pushes B towards right. A goes on compressing the spring till the velocity acquired by B becomes equal to the velocity of A, i.e., 0.15 ms^-1.Let this velocity be v.

Now, spring is in a state of maximum compression. Let x be the maximum compression at this stage.

According to the law of conservation of linear momentum, we get

             ${\mathrm{m}}_{\mathrm{A}} \mathrm{u} = \left({\mathrm{m}}_{\mathrm{A}} + {\mathrm{m}}_{\mathbf{B}}\right) \mathrm{v}$

⇒               v = $\frac{{\mathrm{m}}_{\mathrm{A}} \mathrm{u}}{{\mathrm{m}}_{\mathrm{A}} + {\mathrm{m}}_{\mathrm{B}}}$        

⇒                   = $\frac{2 \times 0.15}{2 + 3}$

                  v = 0.06 ms^-1

According to the law of conservation of energy

         $\frac{1}{2} \mathrm{m}{ }_{\mathrm{A}} {\mathrm{u}}^2 = \frac{1}{2} \left({\mathrm{m}}_{\mathrm{A}} + {\mathrm{m}}_{\mathrm{B}}\right) {\mathrm{v}}^{2 }+ \frac{1}{2} \mathrm{k} {\mathrm{x}}^2$

          $\frac{1}{2} {\mathrm{m}}_{\mathrm{A}} {\mathrm{u}}^{2 }- \frac{1}{2} \left({\mathrm{m}}_{\mathrm{A}} + {\mathrm{m}}_{\mathrm{B}}\right) {\mathrm{v}}^2 = \frac{1}{2} \mathrm{k} {\mathrm{x}}^2$   

   $\frac{1}{2}\times 2 \times {\left(0.15\right)}^2 - \frac{1}{2} \left(2 + 3\right) {\left(0.06\right)}^{2 } = \frac{1}{2 } \mathrm{k} {\mathrm{x}}^2$

                         0.0225 - 0.009 = $\frac{1}{2} \mathrm{k} {\mathrm{x}}^2$

                          0.0135 = $\frac{1}{2} \mathrm{k} {\mathrm{x}}^2$

⇒                              x = $\sqrt{\frac{0.027}{\mathrm{k}}}$

⇒                                 = $\sqrt{\frac{0.027}{10.8}}$

                           x = 0.05 m