| Both sides previous revision Previous revision Next revision | Previous revision |
| 183_notes:springmotion [2015/10/06 10:47] – [Model of a Spring] caballero | 183_notes:springmotion [2024/01/31 17:07] (current) – caballero |
|---|
| | Section 4.10, 4.11 and 4.12 in Matter and Interactions (4th edition) |
| | |
| ===== Non-constant Force: Springs & Spring-like Interactions ===== | ===== Non-constant Force: Springs & Spring-like Interactions ===== |
| |
| In all real-world interactions, the forces acting on a system change with time. This complication is often ignored in physics courses. We often model the motion of systems using constant forces (e.g., the gravitational force near the surface of the Earth) without additional complications (e.g., velocity dependent drag forces). | In all real-world interactions, the forces acting on a system change with time. This complication is often ignored in physics courses. We often model the motion of systems using constant forces (e.g., the gravitational force near the surface of the Earth) without additional complications (e.g., velocity dependent drag forces). |
| |
| However, there are several interactions whose strength and direction depend on the location (and not the velocity) of a system. The simplest of such interactions is the spring force. In these notes, you will read about the spring force, how to determine if the interactions of a system can be modeled using a spring-like force, and how to model the motion of a system subject to spring interaction. | However, there are several interactions whose strength and direction depend on the location (and not the velocity) of a system. The simplest of such interactions is the spring force. **In these notes, you will read about the spring force, how to determine if the interactions of a system can be modeled using a spring-like force, and how to model the motion of a system subject to spring interaction.** |
| |
| ==== Model of a Spring ==== | ===== Model of a Spring ===== |
| <wrap info>Add $\omega$</wrap> | |
| |
| The simulation below shows the interaction of a mass on a spring. | The simulation below shows the interaction of a mass on a spring. |
| |
| {{url>http://www.pa.msu.edu/~caballero/teaching/simulations/HorizontalSpring.html 675px,425px|Mass on a Spring}} | \\ |
| | |
| | |
| | |
| | {{url>https://glowscript.org/#/user/danny/folder/Shared/program/HorizontalSpring 700px,525px|Mass on a Spring}} |
| | |
| | \\ |
| |
| If we plotted the location of this mass (relative to its average location) as a function of time for 10 seconds, we might observe the following: | If we plotted the location of this mass (relative to its average location) as a function of time for 10 seconds, we might observe the following: |
| |
| {{url>https://plot.ly/~dannycab/12/640/480 640px,480px|Plot of spring-mass system}} | \\ |
| | |
| | |
| | |
| | {{url>https://glowscript.org/#/user/danny/folder/Shared/program/SpringMassGraphs 700px,300px|Plot of spring-mass system}} |
| |
| You might have seen this kind of plot before. It's a [[http://en.wikipedia.org/wiki/Sine_wave|sinusoidal function]], in this case it's a sine curve. So the formula that describes this function could be something like: | You might have seen this kind of plot before. It's a [[http://en.wikipedia.org/wiki/Sine_wave|sinusoidal function]], in this case it's a sine curve. So the formula that describes this function could be something like: |
| $$\omega = \dfrac{2\pi}{T}$$ | $$\omega = \dfrac{2\pi}{T}$$ |
| |
| and represents that rate of oscillation in terms of how many radians the sinusoidal functions that models the spring-mass system moves through each second. | __//where $\omega$ is the rate of oscillation in terms of how many radians the spring-mass system moves through each second (rad/s). |
| | //__ |
| |
| [[http://en.wikipedia.org/wiki/Robert_Hooke|Robert Hooke]] investigated this type of motion in the mid 1600s((As it turns out, Hooke was also a great biologist, astronomer, paleontologist, and mathematician. It's amazing what you can do when TV hasn't been invented.)). He found that the interaction (force) that gave rise to this motion was one that increased linearly with the displacement of the object (stretch or compression of the spring) and that was directed opposite the direction of this displacement. In other words, if you stretch a spring (by pulling it), it will pull back on you. If you compress it (by pushing it), it will push back on you. Moreover, these two forces will be equal in size if the stretch and compression are of the same size. | [[http://en.wikipedia.org/wiki/Robert_Hooke|Robert Hooke]] investigated this type of motion in the mid 1600s((As it turns out, Hooke was also a great biologist, astronomer, paleontologist, and mathematician. It's amazing what you can do when TV hasn't been invented.)). He found that the interaction (force) that gave rise to this motion was one that increased linearly with the displacement of the object (stretch or compression of the spring) and that was directed opposite the direction of this displacement. In other words, if you stretch a spring (by pulling it), it will pull back on you. If you compress it (by pushing it), it will push back on you. Moreover, these two forces will be equal in size if the stretch and compression are of the same size. |
| |
| === The spring force is a non-constant force === | ==== The spring force is a non-constant force ==== |
| |
| The force that a spring will exert depends on how far and in what direction it is stretched (or compressed) relative to its //relaxed length.// All springs have a relaxed length where they are neither stretched nor compressed. Mathematically, you can express Hooke's model for the spring force like this: | The force that a spring will exert depends on how far and in what direction it is stretched (or compressed) relative to its //relaxed length.// All springs have a relaxed length where they are neither stretched nor compressed. Mathematically, you can express Hooke's model for the spring force like this: |
| where $L$ is the length of spring (stretched or compressed) and $L_0$ is the relaxed length of the spring. | where $L$ is the length of spring (stretched or compressed) and $L_0$ is the relaxed length of the spring. |
| |
| === Why the minus sign? === | ==== Why the minus sign? ==== |
| |
| In the formula for the spring force, the stretch of the spring is a vector ($\vec{s}$) that points in the direction of the stretch. We observe that the spring force always points in the direction opposite the stretch, so the minus sign in the formula takes care of (and reminds you of) that. | In the formula for the spring force, the stretch of the spring is a vector ($\vec{s}$) that points in the direction of the stretch. We observe that the spring force always points in the direction opposite the stretch, so the minus sign in the formula takes care of (and reminds you of) that. |
| |
| Sometimes, it's useful to first calculate the size of the spring force (i.e., it's magnitude) and then determine the direction in the coordinate system that you have chosen. | Sometimes, it's useful to first calculate the size of the spring force (i.e., it's magnitude) and then determine the direction in the coordinate system that you have chosen. |
| ==== When is an interaction spring-like? ==== | ===== When is an interaction spring-like? ===== |
| |
| The spring force is not a [[http://en.wikipedia.org/wiki/Fundamental_interaction#The_interactions|fundamental force of nature]] ([[183_notes:model_of_solids|as you will learn]]), but rather it has come to be used to describe a class of forces that can modeled using the spring force formula. Any interaction that increases linearly with the displacement of an object and points opposite the direction of that displacement is a spring-like interaction. | The spring force is not a [[http://en.wikipedia.org/wiki/Fundamental_interaction#The_interactions|fundamental force of nature]] ([[183_notes:model_of_solids|as you will learn]]), but rather it has come to be used to describe a class of forces that can modeled using the spring force formula. Any interaction that increases linearly with the displacement of an object and points opposite the direction of that displacement is a spring-like interaction. |
| |
| You can experimentally determine the elastic limit of a spring by hanging successively more weight on it and measuring the stretch. After each measurement, remove the weights and measure the relaxed length of the spring. When that length becomes measurably longer than the initial relaxed length measurement, you can be sure you've reached the elastic limit. | You can experimentally determine the elastic limit of a spring by hanging successively more weight on it and measuring the stretch. After each measurement, remove the weights and measure the relaxed length of the spring. When that length becomes measurably longer than the initial relaxed length measurement, you can be sure you've reached the elastic limit. |
| ==== Modeling Motion with Spring Forces ==== | ===== Modeling Motion with Spring Forces ===== |
| |
| A spring interaction is a non-constant force, which makes systems that experience spring interactions the perfect candidates for [[183_notes:iterativepredict|iterative motion prediction]]. As you might remember, there were 4 steps to predicting motion iteratively: | A spring interaction is a non-constant force, which makes systems that experience spring interactions the perfect candidates for [[183_notes:iterativepredict|iterative motion prediction]]. As you might remember, there were 4 steps to predicting motion iteratively: |
| For the spring force (as well as other non-constant forces), you will use "output" that you obtain about the system from the first 3 steps (i.e., new momentum, new velocity, and new position) as the "input" for the next set of predictions. For the case of spring interactions, you will use the new position of the system to calculate the new spring force. Use the spring force to determine the new momentum, and thus, the new position of the system. Then, repeat. | For the spring force (as well as other non-constant forces), you will use "output" that you obtain about the system from the first 3 steps (i.e., new momentum, new velocity, and new position) as the "input" for the next set of predictions. For the case of spring interactions, you will use the new position of the system to calculate the new spring force. Use the spring force to determine the new momentum, and thus, the new position of the system. Then, repeat. |
| |
| === For predictions, the time step is really important === | ==== For predictions, the time step is really important ==== |
| |
| For [[183_notes:constantf|constant force motion]], the time step ($\Delta t$) was not important because the average velocity ($\vec{v}_{avg} = \dfrac{\Delta \vec{r}}{\Delta t}$) and the //arithmetic// average velocity ($\vec{v}_{avg} = \dfrac{\vec{v}_f + \vec{v}_i}{2}$) were identical. When motion a system results from non-constant interactions, this is no longer true. Consider the figure below, which show predictions of the motion of a spring-mass system using different time steps. Here, you can clearly see that the smaller the time step, the more accurate the plot becomes (i.e., closer to the sinusoidal solution we expect). | For [[183_notes:constantf|constant force motion]], the time step ($\Delta t$) was not important because the average velocity ($\vec{v}_{avg} = \dfrac{\Delta \vec{r}}{\Delta t}$) and the //arithmetic// average velocity ($\vec{v}_{avg} = \dfrac{\vec{v}_f + \vec{v}_i}{2}$) were identical. When motion a system results from non-constant interactions, this is no longer true. Consider the figure below, which show predictions of the motion of a spring-mass system using different time steps. Here, you can clearly see that the smaller the time step, the more accurate the plot becomes (i.e., closer to the sinusoidal solution we expect). |