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A 2.20 kg mass is pushed against a horizontal spring of force constant 25.0 N/cm on a...

Question:

A 2.20 kg mass is pushed against a horizontal spring of force constant 25.0 N/cm on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 15.0 J of potential energy in it, the mass is suddenly released from rest.

a) Find the greatest speed the mass reaches.

b) What is the greatest acceleration of the mass?

Energy Stored by a Spring:

Let a spring of constant k is acted upon by a force such that it got compressed by a distance x . Then the potential energy stored by the spring is given by {eq}E_P= \frac{1}{2}kx^2 {/eq}

Answer and Explanation: 1

Given that a m = 2.20 kg mass is pushed against a horizontal spring of force constant k = 25.0 N/cm =2500N/m on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store E = 15.0 J of potential energy in it, the mass is suddenly released from rest (Initial velocity u= 0).

PART A

The maximum velocity attained by the mass is when the potential energy stored by the spring gets transferred as the kinetic energy of the body

{eq}\begin{align*} E&= \frac{1}{2}mv^2 \\ 15 &=\frac{1}{2} \times 2.2 \times v^2 \\ v&=3.7 m/s \end{align*} {/eq}

PART B

The potential energy stored in the spring when compressed by x is

{eq}\begin{align*} E&=15 J \\ \frac{1}{2}kx^2 &= 15 \\ \frac{1}{2}\times 2500 \times x^2 &= 15 \\ x&=0.11 m \end{align*} {/eq}

The force applied by the spring on the mass is

{eq}\begin{align*} F&=kx\\ &=2500 \times 0.11 N\\ &=273.9N\\\\ \text{Acceleration of the body} &= \frac{Force}{mass}\\ &= \frac{273.9N}{2.2} m/s^2\\ &=124.5 m/s^2 \Rightarrow(Answer) \end{align*} {/eq}


Learn more about this topic:

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Practice Applying Spring Constant Formulas

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Chapter 17 / Lesson 11
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In this lesson, you'll have the chance to practice using the spring constant formula. The lesson includes four problems of medium difficulty involving a variety of real-life applications.


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