I attach a 2.0-kg block to a spring that obeys Hookes Lawand supply 16 J of energy to stretch the...

Question:

I attach a 2.0-kg block to a spring that obeys Hookes Lawand supply 16 J of energy to stretch the spring. I release the block; it oscillates with period 0.30s. The amplitude is

A) 38cm

B) 19cm

C) 9.5cm

D) 4.3cm

Spring:

In a mass-spring system, the amplitude is the distance travelled by the mass after it is released from the sping. The energy stored in the spring due to the compression by an external force is expressed by the following equation:

$$E=\dfrac{1}{2}kx^{2} $$

Answer and Explanation:

We are given the following data:

  • Mass of the block {eq}m=2.0\ \text{kg} {/eq}
  • Energy supplied to stretch the sping {eq}E=16\ \text{J} {/eq}
  • Oscillation period {eq}t=0.30\ \text{s} {/eq}

We are asked to calculate the amplitude of the spring, we can do so by using the following relations:

The energy supplied to compress the spring is expressed by the following equation:

$$E=\dfrac{1}{2}kx^{2} $$

k is the stiffness of the spring

x is the amplitude

The angular velocity of the spring is expressed by the relation:

$$\begin{align} \omega&=\sqrt{\frac{k}{m}}&\left [ m\ \text{is the mass of th block} \right ]\\[0.3 cm] \omega&=\dfrac{2\pi}{t}\\[0.3 cm] \dfrac{2\pi}{t}&=\sqrt{\frac{k}{m}}\\[0.3 cm] \left (\dfrac{2\pi}{t} \right )^{2}&=\dfrac{k}{m}\\[0.3 cm] k&=\left (\dfrac{2\pi}{0.30} \right )^{2}\times2.0\\[0.3 cm] &=877.30\ \text{N/m} \end{align} $$

Calculating the amplitude of the spring from the above relation:

$$\begin{align} E&=\dfrac{1}{2}kx^{2}\\[0.3 cm] 16&=\dfrac{1}{2}\times877.3\times x^{2}\\[0.3 cm] x^{2}&=0.0365\\[0.3 cm] x&=0.190\ \text{m}\\[0.3 cm] &=0.190\times100\ \text{cm}&\left [ \text{1 m=100 cm} \right ]\\[0.3 cm] &=\boxed{19\ \text{cm}} \end{align} $$

So the correct answer is an option (B).


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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