A 2 kg block slides on a horizontal floor with a speed of 4 m / s. It strikes an uncompressed

step 1 Hi student welcome to the solution of problem number 126. 2 kG mass is moving at a velocity of 4

A 2 kg block slides on a horizontal floor with a speed of 4...

A $2 \mathrm{~kg}$ block slides on a horizontal floor with a speed of $4 \mathrm{~m} / \mathrm{s}$. It strikes an uncompressed spring, and compresses it till the block is motionless. The kinetic friction force is $15 \mathrm{~N}$ and spring constant is $10,000 \mathrm{~N} / \mathrm{m}$. The spring compresses by $[\mathbf{2 0 0 7}]$ (A) $8.5 \mathrm{~cm}$ (B) $5.5 \mathrm{~cm}$ (C) $2.5 \mathrm{~cm}$ (D) $11.0 \mathrm{~cm}$

A $2 \mathrm{~kg}$ block slides on a horizontal floor with a speed of $4 \mathrm{~m} / \mathrm{s}$. It strikes an uncompressed spring, and compresses it till the block is motionless. The kinetic friction force is $15 \mathrm{~N}$ and spring constant is $10,000 \mathrm{~N} / \mathrm{m}$. The spring compresses by $[\mathbf{2 0 0 7}]$
(A) $8.5 \mathrm{~cm}$
(B) $5.5 \mathrm{~cm}$
(C) $2.5 \mathrm{~cm}$
(D) $11.0 \mathrm{~cm}$

Key Concepts

Kinetic Energy

Kinetic energy is the energy a mass possesses due to its motion, calculated as one half the mass times the velocity squared. This form of energy is initially present in the system and is transformed into other forms, including stored potential energy and energy dissipated by friction, making it a fundamental starting point in energy transformation problems.

Work Done by Friction

Friction is a non-conservative force that performs work on the system, converting mechanical energy into heat. In dynamics problems where friction is present, it is important to calculate the work done by friction as the product of the frictional force and the distance over which it acts, thereby reducing the energy available for other forms, such as potential energy in a spring.

Hooke's Law and Spring Potential Energy

Hooke's Law describes how the force exerted by a spring is proportional to its displacement from equilibrium, and this relationship is used to determine the potential energy stored in the compressed spring. The potential energy is expressed as one half the product of the spring constant and the square of the displacement, which is a crucial element in problems involving energy conversion in spring systems.

Conservation of Energy with Non-Conservative Forces

This concept involves accounting for energy changes in a system where non-conservative forces, such as friction, are present. The energy initially stored as kinetic energy is not conserved in a simple mechanical form because part of it is dissipated as thermal energy due to friction. Instead, the work-energy theorem is applied, balancing the initial kinetic energy with the energy stored in the spring and the work done against friction.

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