Natural Frequency by Energy Method

Q. When a ball of weight w rests on a spring of constant k, it produces a static deflection of 3 cm. If the ball is now dropped from a height of h = 30 cm as shown in the figure, the spring will get compressed by

1. 2.45 cm
2. 14.07 cm
3. 16.75 cm
4. 33 cm

Q. A block of mass 'm' moving at a speed V compresses a spring through a distance x before its speed is halved. The spring constant of the spring is

1. $\frac{7m{V}^2}{4x^2}$
2. $\frac{3m{V}^2}{4x^2}$
3. $\frac{7m{V}^2}{8x^2}$
4. None of these

Q. Two blocks A and B, each of mass m, are connected by a massless spring of natural length L and spring constant K. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length, as shown in figure. A third identical block C, also of mass m, moves on the floor with a speed v along the line joining A and B, and collides elastically with A.

Consider the following statements. Which of the following is/are correct?

(i) The kinetic energy of the A-B system, at maximum compression of the spring, is zero

(ii) The kinetic energy of the A- B system , at maximum compression of the spring, is $\frac{m{v}^2}{4}$

(iii) The maximum compression of the spring, is $v\sqrt{\left(\frac{m}{K}\right)}$

(iv) The maximum compression of the spring, is $v\sqrt{\left(\frac{m}{2K}\right)}$

1. (ii) & (iv) only
2. (i) only
3. (ii) & (iii) only
4. (i) & (iii) only