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183_notes:examples:calculating_the_force_due_to_a_stretched_spring [2014/07/20 06:30] – pwirving | 183_notes:examples:calculating_the_force_due_to_a_stretched_spring [2014/07/22 04:55] (current) – 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 $N/m$ and 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 |
=== Facts ==== | === Facts ==== | ||
- | | + | |
- | * Spring has spring constant of $8 N/m$ | + | |
- | * 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 interest, the 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}$ |
- | | + | $ |\vec{s}| = |L - L_0|$ |
- | {{183_notes: | + | {{183_notes: |
- | ==== Solution ==== | + | {{183_notes: |
- | | ||
- | $|\vec{L}| = 0.38m$ | + | ==== Solution ==== |
+ | |||
+ | To determine the spring force, you will need to compute: | ||
+ | $$ {\vec F_{spring}} = -k_s\vec{s} = -k_s|\vec{s}|\hat{s}$$ | ||
- | $\hat{L} = \dfrac{(0.38,0,0)}{0.38} = \langle | + | You will start be determining the position vector ($\vec{L}$) of the mass and the length of the position vector ($|\vec{L}|$), |
+ | | ||
- | | + | $$|\vec{L}| |
- | $\vec{F} = -(8N/m)(0.18m)(1,0,0) = \langle 1.44,0,0 \rangle | + | These can be used to compute the unit (direction) vector for the stretch ($\hat{s}$), which is in the same direction as the position vector: |
+ | | ||
+ | You can then compute the magnitude of the stretch $(|\vec{s}|)$: | ||
+ | $$ |\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/ | ||
+ | which points to the left. That is consistent with the diagram above. | ||