Editing Talk:2904: Physics vs. Magic
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The relation (5) is known as Torricelli's formula. <br> | The relation (5) is known as Torricelli's formula. <br> | ||
− | In case the initial position coordinate and the initial velocity are non-zero | + | In case the initial position coordinate and the initial velocity are non-zero: |
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− | + | a(s-s₀) = ½v² - ½(v₀)² (6) | |
− | + | Interestingly, in the case of a non-uniform acceleration the result of the integration is identical to (6) | |
− | (To understand ( | + | ∫ a ds = ½v² - ½(v₀)² (7) |
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+ | (To understand (7): we have that integration is summation of infinitesimal strips. The integration consists of concatenating instances of (6), in the limit of infinitesimal increments. All of the in-between terms drop away against each other, resulting in (7)) | ||
<br> | <br> | ||
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The work-energy theorem is obtained as follows: start with ''F''=''ma'', and integrate both sides with respect to the position coordinate. | The work-energy theorem is obtained as follows: start with ''F''=''ma'', and integrate both sides with respect to the position coordinate. | ||
− | ∫ F ds = ∫ ma ds ( | + | ∫ F ds = ∫ ma ds (8) |
− | Use ( | + | Use (7) to process the right hand side: |
− | ∫ F ds = ½mv² - ½m(v₀)² ( | + | ∫ F ds = ½mv² - ½m(v₀)² (9) |
− | ( | + | (9) is the work-energy theorem |
The work-energy theorem is the reason that it is useful to formulate the concepts of potential energy and kinetic energy. If we formulate potential energy and kinetic energy in accordance with the work-energy theorem then we have that the sum of potential energy and kinetic energy is a conserved quantity. | The work-energy theorem is the reason that it is useful to formulate the concepts of potential energy and kinetic energy. If we formulate potential energy and kinetic energy in accordance with the work-energy theorem then we have that the sum of potential energy and kinetic energy is a conserved quantity. | ||
− | The work-energy theorem consists of two elements: ''F''=''ma'', and ( | + | The work-energy theorem consists of two elements: ''F''=''ma'', and (7). |
− | Here ( | + | Here (7) was stated in terms of the familiar quantities of mechanics: position, velocity, acceleration. (7) generalizes to any set of three quantites that features that cascading relation: state, first time derivative, second time derivative. |
Example: electric current and electromotive force in an LC circuit<br> | Example: electric current and electromotive force in an LC circuit<br> | ||
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Conservation laws and Lagrangian refer to their role in particle physics. For example, conservation of electrical charge is simply postulated as something that your theory should satisfy, but physics does not tell us why it is there in the first place (hence: magic). Similarly, Lagrangians are usually formulated in a way that the outcomes are compatible with experimental observations and not starting out based on fundamental principles. The statement "Lagrangian mechanics instead takes the initial and final states of a system as inputs" is wrong by the way. The Euler-Lagrange equation yields a differential equation that is usually solved as an initial value problem, as it is done in Newtonian mechanics. {{unsigned ip|172.70.110.144|07:38, 11 March 2024}} | Conservation laws and Lagrangian refer to their role in particle physics. For example, conservation of electrical charge is simply postulated as something that your theory should satisfy, but physics does not tell us why it is there in the first place (hence: magic). Similarly, Lagrangians are usually formulated in a way that the outcomes are compatible with experimental observations and not starting out based on fundamental principles. The statement "Lagrangian mechanics instead takes the initial and final states of a system as inputs" is wrong by the way. The Euler-Lagrange equation yields a differential equation that is usually solved as an initial value problem, as it is done in Newtonian mechanics. {{unsigned ip|172.70.110.144|07:38, 11 March 2024}} | ||
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