Figure (8-E12) shows two blocks A and B, each having a mass of 320 g connected by a light strin

step 1 So the problem goes like this. There is a block A, which is kept on a smooth surface, and it is

Figure (8-E12) shows two blocks A and B, each having a mass...

Figure (8-E12) shows two blocks $A$ and $B$, each having a mass of $320 \mathrm{~g}$ connected by a light string passing over a smooth light pulley. The horizontal surface on which the block $A$ can slide is smooth. The block $A$ is attached to a spring of spring constant $40 \mathrm{~N} / \mathrm{m}$ whose other end is fixed to a support $40 \mathrm{~cm}$ above the horizontal surface. Initially, the spring is vertical and unstretched when the system is released to move. Find the velocity of the block $A$ at the instant it breaks off the surface below it. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$

Key Concepts

Hooke’s Law and Elastic Potential Energy

Hooke’s Law describes the force exerted by a spring when it is stretched or compressed, which is proportional to the displacement from its equilibrium position. The elastic potential energy stored in the spring is a key factor in dynamics problems involving springs, as it can be converted into kinetic energy or other forms of potential energy as the system moves.

Constraints of Connected Bodies

When two objects are connected by an inextensible string over a smooth pulley, their motions are interdependent. This means that the displacement, velocity, and acceleration of one block are directly related to those of the other. Recognizing these constraints allows one to set up equations that couple the motion of the different bodies involved.

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (like friction), the total mechanical energy (the sum of kinetic and potential energies) of a system remains constant. In problems involving masses, springs, and gravity, energy conservation is used to equate the loss of gravitational potential energy and changes in elastic potential energy to the gain in kinetic energy.

Ideal Pulley and Frictionless Surface Assumptions

Assuming an ideal (smooth and massless) pulley and a frictionless surface simplifies the dynamics by eliminating energy losses due to friction and rotational inertia. These assumptions ensure that all energy changes in the system are accounted for purely by the gravitational, elastic, and kinetic energies, allowing straightforward application of conservation principles.

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