  # Oscillations Due to a Spring

Physics

## example

### Problem on velocity and amplitude of mass attched to spring

Example: A block with mass is connected by a mass less spring with stiffness constant to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position . Consider two cases: (i) when the block is at ; and (ii) when the block is at . In both the cases, a particle with mass is softly placed on the block after which they stick to each other.
Which of the following statement(s) is(are) true about the motion after the mass is placed on the mass ?

Solution:
In case I,From Conservation of momentum,

In case II,

in both the cases.
Total energy decreases in first case where as remain same in 2nd case.
Instantaneous speed at decreases in both case.

## example

### Write force equations of an arrangement of springs in parallel and calculate equivalent spring constant

Example: Two blocks and attached to the ends of a spring constant 1120N/m are placed on a smooth horizontal plane with the spring undeformed. Simultaneously, velocities of 3m/s and 10m/s along the line of the spring in the same direction are imparted to and , then find the maximum extension of the spring.

Solution:
The maximum elongation occurs when the velocity of both the particles becomes equal. Since there is no external force acting in the horizontal direction (for the
blocks + spring system), thus the linear momentum is conserved in horizontal direction:
If the final common velocity is

Also applying law of conservation of energy for the system (work done by non-conservative force is zero):
Initial Energy Final Energy

Solving the above equation we get cm
Also it can be observed that the compression and expansion occur alternately.