Iterative Prediction of Spring-Mass System: Difference between revisions
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The simplest example of a spring mass system is one that moves in only one-direction. Consider a massless spring of length 1.0 m with spring constant 100 N/m. If a 1 kg mass is released from rest while the spring is stretched downward to a length of 1.5 m, what is it's momentum after 0.5 seconds? The mass oscillates vertically, as shown to the right. | The simplest example of a spring mass system is one that moves in only one-direction. Consider a massless spring of length 1.0 m with spring constant 100 N/m. If a 1 kg mass is released from rest while the spring is stretched downward to a length of 1.5 m, what is it's momentum after 0.5 seconds? The mass oscillates vertically, as shown to the right. | ||
''Step 1: Calculate Initial Values | ''Step 1: Calculate Initial Values'' Begin by calculating the object's initial momentum and the sum of the forces acting on it. The initial momentum is simply the product of the initial velocity, which is 0 m/s, as the object is released ''from rest''. The forces acting on the mass are the gravitational force exerted by the Earth and the tension exerted by the spring. F_net can be calculated by summing these two forces. | ||
<math>{\vec{p}_{i}} = {{m}\cdot\vec{v}_{i}}</math> | <math>{\vec{p}_{i}} = {{m}\cdot\vec{v}_{i}}</math> | ||
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''Step 2: Update Momentum'' | |||
Using the momentum update formula, calculate the momentum of the mass at the end of the given time step. | |||
<math>{\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}}</math> | |||
<math>{\vec{p}_{f} = {0Ns} + {40.2N}\cdot{0.1s} = {4.2Ns}}</math> | |||
Revision as of 17:11, 1 December 2015
claimed by kgiles7
Short Description of Topic
The Main Idea
A simple spring-mass system is a basic illustration of the momentum principle. The principle of conservation of momentum can be repeatedly applied to predict the system's future motion.
A Mathematical Model
The Momentum Principle provides a mathematical basis for the repeated calculations needed to predicts the system's future motion. The most useful form of this equation for predicting future motion is referred to as the momentum update form, and can be derived by rearranging the Momentum Principle as shown below:
[math]\displaystyle{ {\vec{p}_{f} - \vec{p}_{i} = \vec{F}_{net}{Δt}} }[/math]
[math]\displaystyle{ {\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}} }[/math]
Velocity Update Formula: [math]\displaystyle{ {\vec{v}_{f} = \vec{v}_{i} + \frac{\vec{F}_{net}}{m}}{Δt} }[/math]
Position Update Formula: [math]\displaystyle{ {\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{avg}{Δt}} }[/math]
A Computational Model
How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript
Examples
Be sure to show all steps in your solution and include diagrams whenever possible
Simple Example
The simplest example of a spring mass system is one that moves in only one-direction. Consider a massless spring of length 1.0 m with spring constant 100 N/m. If a 1 kg mass is released from rest while the spring is stretched downward to a length of 1.5 m, what is it's momentum after 0.5 seconds? The mass oscillates vertically, as shown to the right.
Step 1: Calculate Initial Values Begin by calculating the object's initial momentum and the sum of the forces acting on it. The initial momentum is simply the product of the initial velocity, which is 0 m/s, as the object is released from rest. The forces acting on the mass are the gravitational force exerted by the Earth and the tension exerted by the spring. F_net can be calculated by summing these two forces.
[math]\displaystyle{ {\vec{p}_{i}} = {{m}\cdot\vec{v}_{i}} }[/math]
[math]\displaystyle{ {\vec{p}_{i}} = {{10kg}\cdot{0m/s}} = {0 Ns} }[/math]
[math]\displaystyle{ {\vec{F}_{grav}} = {mg} }[/math]
[math]\displaystyle{ {\vec{F}_{grav}} = {10kg}\cdot{-9.8 m/s/s} = {-9.8 N} }[/math]
[math]\displaystyle{ {\vec{F}_{spring}} = {-kx} }[/math]
[math]\displaystyle{ {\vec{F}_{spring}} = {-100N/m}\cdot{1.0m-1.5m} = {50 N} }[/math]
[math]\displaystyle{ {\vec{F}_{net}} = {\vec{F}_{grav} +{\vec{F}_{spring}}} }[/math]
[math]\displaystyle{ {\vec{F}_{net}} = {-9.8N} +{50 N} = {40.2N} }[/math]
Step 2: Update Momentum
Using the momentum update formula, calculate the momentum of the mass at the end of the given time step.
[math]\displaystyle{ {\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}{Δt}} }[/math]
[math]\displaystyle{ {\vec{p}_{f} = {0Ns} + {40.2N}\cdot{0.1s} = {4.2Ns}} }[/math]
The time step used for each iteration must be small enough that we can assume a constant velocity over the interval, but not so large that solving the problem becomes incredibly time-consuming. An appropriately small time step here is approximately 0.1 seconds.
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