Order of anharmonicity of a simple pendulum

In summary, the potential of a simple pendulum can be calculated using the formula ##V=mgl\left(\frac{\theta^2}{2}-\frac{\theta^4}{24}+...\right )## where ##\theta## is the angle of displacement. The anharmonicity in the potential is given by ##O(x^3)##, which is the third order term in the expansion of ##cos\theta##. This is different from the anharmonicity in the force, which is ##O(x^3)##. The potential is symmetric about the equilibrium position and does not contain any odd terms in its expansion.
  • #1
Saptarshi Sarkar
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Homework Statement
The anharmonicity in the potential of a simple pendulum is of the order of

1. ##\theta##
2. ##\theta^2##
3. ##\theta^3##
4. ##\theta^4##
Relevant Equations
##V=mgl(1-cos\theta)##
I know that the potential of a simple pendulum is given by the above formula and that we can expand ##cos\theta## to get

##V=mgl\left(\frac{\theta^2}{2}-\frac{\theta^4}{24}+...\right )##

I am guessing that the answer is ##\theta^4##, but I am not sure what "order" means here.
 
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  • #2
That would be my guess too.
 
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  • #3
It seems at least as valid to say that the standard harmonic equation is ##\ddot x=-k^2x##, but the pendulum is ##\ddot x=-k^2\sin(x)=-k^2x+O(x^3)##, so the anharmonicity is ##O(x^3)##.
 
  • #4
haruspex said:
It seems at least as valid to say that the standard harmonic equation is ##\ddot x=-k^2x##, but the pendulum is ##\ddot x=-k^2\sin(x)=-k^2x+O(x^3)##, so the anharmonicity is ##O(x^3)##.
Yes, but that is the anharmonicity in the force not the potential which is what the question asks. The potential is symmetric about the equilibrium position and has no odd terms in its expansion.
 
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  • #5
kuruman said:
Yes, but that is the anharmonicity in the force not the potential which is what the question asks. The potential is symmetric about the equilibrium position and has no odd terms in its expansion.
Thanks, I missed that it specified potential.
 

Related to Order of anharmonicity of a simple pendulum

1. What is the order of anharmonicity of a simple pendulum?

The order of anharmonicity of a simple pendulum refers to the degree of nonlinearity in the pendulum's motion. It is typically denoted by the symbol α and can range from 0 (harmonic motion) to 1 (highly anharmonic motion).

2. How is the order of anharmonicity determined?

The order of anharmonicity is determined by analyzing the pendulum's motion and comparing it to the ideal harmonic motion. This can be done through mathematical calculations or by plotting the pendulum's displacement over time and observing any deviations from a sinusoidal curve.

3. What factors can affect the order of anharmonicity?

The order of anharmonicity can be affected by various factors such as the length and mass of the pendulum, the amplitude of its oscillations, and the strength of the restoring force (e.g. gravity). In general, a longer and heavier pendulum with larger amplitudes will exhibit higher levels of anharmonicity.

4. How does the order of anharmonicity impact the accuracy of a simple pendulum?

The order of anharmonicity can have a significant impact on the accuracy of a simple pendulum. As the pendulum becomes more anharmonic, its motion deviates further from the ideal harmonic motion, making it more difficult to predict its behavior and measure its period accurately.

5. Can the order of anharmonicity be reduced in a simple pendulum?

Yes, the order of anharmonicity can be reduced in a simple pendulum by minimizing the factors that contribute to its nonlinearity. For example, using a lighter and shorter pendulum with smaller amplitudes can help reduce the order of anharmonicity and improve the accuracy of its measurements.

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