Two blocks A and B of equal mass m=1 kg are lying on a smooth horizontal surface as shown in Fi

step 1 In this question we have given two blocks A and B both have equal masses and lying on a smooth h

Two blocks A and B of equal mass m=1 kg are lying on a...

Two blocks $A$ and $B$ of equal mass $m=1 \mathrm{~kg}$ are lying on a smooth horizontal surface as shown in Fig. $4.27$. A spring of force constant $K=200 \mathrm{~N} / \mathrm{m}$ is fixed at one end of block $A$. Block $B$ collides with another end of the spring with velocity $v_{0}=2 \mathrm{~m} / \mathrm{s}$. What will be the maximum compression of the spring? [in decimeter $]$

Key Concepts

Conservation of Momentum

In any isolated system where external forces are negligible, the total momentum remains constant throughout the process. In collision problems, especially those involving multiple objects interacting through internal forces like springs, the momentum before the interaction must equal the momentum after, which is critical when analyzing the transfer of motion between objects.

Conservation of Mechanical Energy

In frictionless scenarios, mechanical energy is conserved, meaning that the initial kinetic energy of the system is transformed into other forms of mechanical energy, such as the potential energy stored in a compressed spring. This principle is essential for determining how much energy is available to compress the spring at the moment when kinetic energy is minimized.

Spring Potential Energy

A spring stores energy when it is compressed or extended, and this stored energy is given by (1/2)Kx², where K is the spring constant and x is the displacement from the equilibrium position. Recognizing this form of potential energy is crucial for equating the energy lost by the moving objects to the energy gained in compressing the spring.

Equal Velocity Condition at Maximum Compression

In scenarios involving a spring and colliding masses, the maximum compression of the spring occurs when the two blocks momentarily reach the same velocity. This condition implies that, at maximum compression, the relative motion between the blocks is zero, meaning that the kinetic energy has been optimally converted into spring potential energy. This equality of velocities is fundamental to setting up the energy conservation equations needed to solve the problem.

Transcript

Please give Ace some feedback