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If a known mass is suspended vertically from a spring, and you have means of measuring distance...

Question:

If a known mass is suspended vertically from a spring, and you have means of measuring distance and time information, determine some ways to measure the unknown spring constant of the spring. State at least two ways and include equations to support the approach.

Spring Constant:

The spring constant is a constant that is a consequence of Hooke's law, which governs spring-mass motion problems. According to Hooke's law,

{eq}\displaystyle F = -kx {/eq}

where x is the compression distance of the spring.

Moreover, the period of oscillation can be determined as:

{eq}\displaystyle T = \sqrt{\frac{m}{k}} {/eq}

where m is the mass of the hanging object.

Answer and Explanation:

(a) For the first method, we will use:

{eq}\displaystyle F = -kx {/eq}

with Newton's laws of force and motion. We know that the mass is hanging vertically from the spring. In this case, we should wait for a moment until it stops moving and is just hanging there. Now Newton's laws can be applied, which state that forces can add up to a total force:

{eq}\displaystyle F = ma {/eq}

Since the object is hanging vertically, there will also be a gravitational force, g, acting on it. This can be inserted into the force equation to yield:

{eq}\displaystyle ma = mg - kx {/eq}

Since the compression distance is upward, then the gravitational force always has to act oppositely. Since the object is not moving, then:

{eq}\displaystyle ma = 0 {/eq}

And:

{eq}\displaystyle 0 = mg - kx {/eq}

We isolate k here to obtain:

{eq}\displaystyle k = \frac{mg}{x} {/eq}

where:

  • m is the mass of the object
  • x is the length of the spring (not compressed)
  • {eq}\displaystyle g = 9.81\ m/s^2 {/eq} is the gravitational constant.

(b) For the second method, we can use the period of oscillation. In this case, we attempt to move the object. It will now oscillate, as it is still attached to the spring. We can determine the distance between the oscillation endpoints as well as the time it takes to reach these distances. In this case,

{eq}\displaystyle T = \sqrt{\frac{m}{k}} {/eq}

Here, the period of oscillation is the time it takes for the mass to return to its original position after oscillating one full period. We square both sides,

{eq}\displaystyle T^2 = \frac{m}{k} {/eq}

And isolate the spring constant:

{eq}\displaystyle k = \frac{m}{T^2} {/eq}

where:

  • m is the mass of the object
  • T is the period of one full oscillation

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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