Tendons are strong elastic fibers that attach muscles to bones. To a reasonable approximation,...

Question:

Tendons are strong elastic fibers that attach muscles to bones. To a reasonable approximation, they obey Hooke's law. In laboratory tests on a particular tendon, it was found that when a 249 g object was hung from it, the tendon stretched 1.25 cm.

a. Find the force constant of this tendon in N/m.
b. Because of its thickness, the maximum tension this tendon can support without rupturing is 143 N. By how much can the tendon stretch without rupturing?

Force Constant:


The force constant of any elastic object (rope, string, etc.) denotes its strength against a deflection-producing force. This force constant is a measure that reveals how stiff the object is or how much force is required to produce one-meter deflection in it.

Answer and Explanation: 1


We are given the following data:

  • The mass of the object is {eq}{m_o} = 249\;{\rm{g}} = 0.249\;{\rm{kg}} {/eq}.
  • The initial stretch in the tendon is {eq}{x_i} = 1.25\;{\rm{cm}} = 0.0125\;{\rm{m}} {/eq}.
  • The maximum tension that can be supported by the tendon is {eq}{T_m} = 143\;{\rm{N}} {/eq}.


Question (a)


The expression to calculate the force constant of the tendon is as follows:

{eq}s{x_i} = {m_o}g {/eq}

{eq}s = \dfrac{{{m_o}g}}{{{x_i}}} {/eq}, where:

  • {eq}g {/eq} is the acceleration due to gravity.


Substituting all the known values into the above expression, we have:

{eq}\begin{align*} s &= \dfrac{{0.249 \times 9.81}}{{0.0125}}\\ & = 1.95\times10^2\;{\rm{N/m}} \end{align*} {/eq}


Thus, the force constant of the tendon is {eq}\mathbf{1.95\times10^2\;{N/m}} {/eq}.


Question (b)


The expression to calculate the maximum stretch in the tendon without rupture is as follows:

{eq}\begin{align*} {T_m} &= s{x_m}\\ {x_m} &= \dfrac{{{T_m}}}{s} \end{align*} {/eq}


Substituting all the values into the above expression, we have:

{eq}\begin{align*} {x_m} &= \dfrac{{143}}{{1.95\times10^2}}\\ & \approx 0.733\;{\rm{m}}\\ & \approx 0.733\;{\rm{m}} \times \left( {\dfrac{{100\;{\rm{cm}}}}{{1\;{\rm{m}}}}} \right)\\ & \approx 73.3\;{\rm{cm}} \end{align*} {/eq}


Thus, the maximum stretch in the tendon without rupture is 73.3 cm.


Learn more about this topic:

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Hooke's Law & the Spring Constant: Definition & Equation

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Chapter 4 / Lesson 19
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After watching this video, you will be able to explain what Hooke's Law is and use the equation for Hooke's Law to solve problems. A short quiz will follow.


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