In Fig. 8-53, a block of mass m=2.5 kg slides head on into a spring of spring constant k=320 N

step 1 Well, we have mass of block and it's equal to 2 .5 kilograms. So mass of block is equal to M and

In Fig. 8-53, a block of mass m=2.5 kg slides head on into...

In Fig. 8-53, a block of mass $m=2.5 \mathrm{~kg}$ slides head on into a spring of spring constant $k=320 \mathrm{~N} / \mathrm{m}$. When the block stops, it has compressed the spring by $7.5 \mathrm{~cm}$. The coefficient of kinetic friction between block and floor is $0.25$. While the block is in contact with the spring and being brought to rest, what are (a) the work done by the spring force and (b) the increase in thermal energy of the block-floor system? (c) What is the block's speed just as it reaches the spring?

In Fig. 8-53, a block of mass $m=2.5 \mathrm{~kg}$ slides head on into a spring of spring constant $k=320 \mathrm{~N} / \mathrm{m}$. When the block stops, it has compressed the spring by $7.5 \mathrm{~cm}$. The coefficient
of kinetic friction between block and floor is $0.25$. While the block is in contact with the spring and being brought to rest, what are (a) the work done by the spring force and (b) the increase in thermal energy of the block-floor system? (c) What is the block's speed just as it reaches the spring?

Key Concepts

Work-Energy Theorem

The work-energy theorem states that the net work done on an object equals its change in kinetic energy. This theorem serves as a foundational principle connecting forces (and the work they do) with changes in the energy state of a system, allowing for analysis of energy transformations when non-conservative forces, like friction or spring forces, are present.

Hooke's Law and Elastic Potential Energy

This concept explains that a spring, when compressed or stretched, stores energy in the form of elastic potential energy. The amount of energy stored is given by the expression ½kx², where k is the spring constant and x is the compression or extension. It is crucial for understanding how mechanical energy is temporarily held in the spring during interactions such as collisions.

Friction and Thermal Energy

This concept involves kinetic friction, which is a non-conservative force that opposes motion. When friction acts on a moving object, it does negative work, transforming some of the mechanical energy of the object into thermal energy. Understanding this conversion is key for analyzing energy dissipation in real-world systems.

Conservation of Energy

In mechanical systems, the principle of conservation of energy dictates that the total energy remains constant unless it is altered by non-conservative forces. In scenarios involving springs and friction, the initial mechanical energy of a system is redistributed between elastic potential energy and converted thermal energy, highlighting the importance of energy accounting in dynamic processes.

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