183_notes:examples:calculating_the_force_due_to_a_stretched_spring

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183_notes:examples:calculating_the_force_due_to_a_stretched_spring [2014/07/20 06:29] pwirving183_notes:examples:calculating_the_force_due_to_a_stretched_spring [2014/07/22 04:54] pwirving
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 ===== Example: Calculating the force due to a stretched spring ===== ===== Example: Calculating the force due to a stretched spring =====
  
-A spring with a mass block at the end of it and with a stiffness of 8N/m and a relaxed length of 20cm is attached to a chamber wall that results in its oscillations being horizontal. At a particular time the location of the block mass is $\langle .38,0,0 \rangle$ relative to an origin point where the spring is attached to the chamber wall. What is the force exerted by the spring on the mass at this instant?+A spring with a mass block at the end of it and with a stiffness of 8 $N/mand a relaxed length of 20 $cm$ is attached to a chamber wall that results in its oscillations being horizontal. At a particular time the location of the block mass is $\langle .38,0,0 \rangle\,m$ relative to an origin point where the spring is attached to the chamber wall. Determine the force exerted by the spring on the mass at this instant.
  
  
 === Facts ==== === Facts ====
-  * Spring has relaxed length of (0.2m$L_0=0.2m+ 
-  * Spring has spring constant of $8 N/m$ +  * Spring has relaxed length of 0.2m$L_0=0.2\,m
-  * At the moment of interest the mass block is at position $\vec{L} = \langle .38,0,0 \rangle m$ +  * Spring has spring constant of $8 N/m$, $k_s=8\,N/m$ 
-  * Only force acting on system is spring force+  * At the moment of interestthe mass block is at position $\vec{L} = \langle .38,0,0 \rangle m$ 
 +  * The net force acting on system is due to spring force (the gravitational force exerted by the Earth has the same magnitude as the force exerted by the horizontal surface)
  
 === Lacking === === Lacking ===
  
 +  * The force that the spring exerts
  
 === Approximations & Assumptions === === Approximations & Assumptions ===
-  * Origin is at chamber wall $\langle 0,0,0 \rangle$+  * Origin is at chamber wall $\langle 0,0,0 \rangle\,m$
   * Assume no forces due to drag or to friction   * Assume no forces due to drag or to friction
      
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 === Representations === === Representations ===
    
- $ {\vec F_{spring}} = -k_ss\hat{L}$+ $ {\vec F_{spring}} = -k_s\vec{s}$
    
- s = |\vec L| - L_0$+ $ |\vec{s}= |L - L_0|$
  
-{{183_notes:spring_force_jpeg.jpg}}+{{183_notes:spring237.jpg}}
  
-==== Solution ====+{{183_notes:spring_235.jpg}}
  
- $\vec{L} = \langle 0.38,0,0 \rangle m - \langle 0,0,0 \rangle m = \langle 0.38,0,0 \rangle m$+<WRAP todo>This FBD needs the forces of the Earth and the surfaceIt should also be abstractFor examplesee the early fan cart problems.</WRAP> 
 +==== Solution ====
  
- $|\vec{L}| = 0.38m$+To determine the spring force, you will need to compute: 
 +$$  {\vec F_{spring}} = -k_s\vec{s} = -k_s|\vec{s}|\hat{s}$$
  
- $\hat{L} \dfrac{(0.38,0,0)}{0.38} = \langle 1,0,0 \rangle m$+You will start be determining the position vector ($\vec{L}$) of the mass and the length of the position vector ($|\vec{L}|$), 
 + $$\vec{L} = \langle 0.38,0,\rangle m - \langle 0,0,0 \rangle m = \langle 0.38,0,0 \rangle m$$
  
- $= 0.38m - 0.20m = 0.18m+ $$|\vec{L}| = 0.38m$$
  
- $\vec{F= -(8N/m)(0.18m)(1,0,0= \langle 1.44,0,0 \rangle N/m$+These can be used to compute the unit (direction) vector for the stretch ($\hat{s}$), which is in the same direction as the position vector: 
 + $$\hat{s} = \hat{L} = \dfrac{\langle 0.38,0,0\rangle}{0.38} = \langle 1,0,0 \rangle$$
  
 +You can then compute the magnitude of the stretch $(|\vec{s}|)$:
 + $$ |\vec{s}| = |L - L_0| = 0.38m - 0.20m = 0.18m$$
  
 +Finally, you can compute the force:
  
 +$$\vec{F} = -k_s|\vec{s}|\hat{s} = -(8N/m)(0.18m)\langle 1,0,0\rangle = \langle -1.44,0,0 \rangle\,N$$
  
 +which points to the left. That is consistent with the diagram above.
  
  
  • 183_notes/examples/calculating_the_force_due_to_a_stretched_spring.txt
  • Last modified: 2014/07/22 04:55
  • by pwirving