### Problem StatementSuppose you fire a mass \( m \) straight vertically upwards from a spring that is known to have been compressed a distance \( x \) from its equilibrium position, and the mass reaches a maximum height of \( h \) above its launch point. Using the conservation of energy, derive an expression for the spring constant \( k \) of the spring.### Educational ExplanationTo solve this problem, we apply the principle of conservation of energy, which states that the total mechanical energy in the system remains constant if only conservative forces act on it.#### 1. Initial EnergyInitially, the spring is compressed by a distance \( x \). The potential energy stored in the spring, known as elastic potential energy, can be expressed as:\[ E_{\text{spring}} = \frac{1}{2}kx^2 \]where \( k \) is the spring constant.#### 2. Energy at Maximum HeightAt the maximum height \( h \), all the initial spring potential energy is converted into gravitational potential energy. The gravitational potential energy is given by:\[ E_{\text{gravity}} = mgh \]where \( g \) is the acceleration due to gravity.#### 3. Conservation of EnergyAccording to the conservation of energy:\[ \frac{1}{2}kx^2 = mgh \]#### 4. Deriving the Spring ConstantWe need to solve for \( k \). Rearranging the equation:\[ k = \frac{2mgh}{x^2} \]### ConclusionThus, by using the conservation of energy, we derived the expression for the spring constant:\[ k = \frac{2mgh}{x^2} \]This expression shows how the spring constant is related to the mass, gravitational acceleration, maximum height, and compression distance.

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Description: ### Problem StatementSuppose you fire a mass \( m \) straight vertically upwards from a spring that is known to have been compressed a distance \( x \) from its equilibrium position, and the mass reaches a maximum height of \( h \) above its launch

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uppose you fire a mass m straight vertically upwards from a spring that is knc compressed a distance x from its equilibrium position, and the mass reac eight of h above its launch point. Using the conservation of energy, derive

uppose you fire a mass m straight vertically upwards from a spring that is knc compressed a distance x from its equilibrium position, and the mass reac eight of h above its launch point. Using the conservation of energy, derive

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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