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1. Top results related to parallel axis theorem

   • Parallel Axis Theorem: Definition, History, Formula, Proof

     Parallel axis theorem states that the moment of inertia of a rigid body about any axis parallel to its centroidal axis is equal to the sum of the body’s moment of inertia about its centroidal axis and the product of its mass and the square of the distance between the two axes.

     In this article, we will understand the meaning of the parallel axis theorem, its history, proof, limitations and applications of parallel axis theorem.

     Table of Content

     • What is the Parallel Axis Theorem?
     • What is the Formula of the Theorem of Parallel Axis?
     • Parallel Axis Theorem Proof
     • Limitations of Parallel Axis Theorem
     • Application of Parallel Axis Theorem

     What is Parallel Axis Theorem?According to Parallel Axis Theorem, the moment of inertia of a body about any axis is equal to the sum of following two components:

     • Product of the body’s mass and the square of the distance between the axis and the body’s center of mass,
     • Moment of inertia of the body about a parallel axis passing through its center of mass.

     History of the Parallel Axis Theorem The concept of the moment of inertia was first introduced by the polymath Christiaan Huygens while studying a compound pendulum. Later, Jakob Steiner made significant contributions to this theorem. In recognition of their work, the parallel axis theorem is often referred to as the Huygens-Steiner theorem. This theorem provides a method for calculating the moment of inertia of a rigid body about an axis parallel to an axis passing through its center of mass....

     • I is the moment of inertia about a parallel axis,
     • Icm​ is the moment of inertia about the center of mass (also known as the centroid),
     • m is the mass of the object, and
     • d is the perpendicular distance between the two axes.
     • The theorem relies on the assumption that the mass distribution within the object remains constant. If this assumption is violated, the theorem’s accuracy diminishes.
     • It is valid only for rigid bodies, where the distance between the parallel axes remains fixed. For objects experiencing significant deformations, its applicability may be limited.
     • While it can be extended to three-dimensional objects, calculations become more complex, potentially reducing accuracy.
     • External forces acting on the object are not considered in the theorem, which may lead to inaccuracies in dynamic scenarios.
     • It primarily addresses linear motion and may not adequately describe rotational motion in all cases, especially non-linear or chaotic motion.
     • Rotational Dynamics: The theorem is widely used in rotational dynamics to calculate the moment of inertia of objects about axes parallel to their centroidal axis. This facilitates the analysis of rotational motion in various engineering and physics applications.
     • Engineering Design: Engineers use he theorem in designing machinery, vehicles, and structures, where understanding rotational motion is critical. It aids in optimizing designs for stability, efficiency, and performance.
     • Aerospace Industry: In aerospace engineering, the theorem helps in designing aircraft components, such as wings, propellers, and control surfaces. Accurate moment of inertia calculations are crucial for flight dynamics and stability.
     • Robotics: Parallel axis theorem is employed in robotics for designing robotic arms, joints, and end-effectors. It enables engineers to determine the required torque and motor specifications for precise and efficient movement.
     • Materials Science: In materials science, the theorem is applied to analyze the mechanical properties of materials, such as stiffness and strength. Understanding the moment of inertia aids in predicting the behavior of materials under various loading conditions.
     • Rotational Motion
     • Center of Gravity
     • Radius of Gyration

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   • Parallel axis dot plot with vertical histograms for paired samples

     Top Answer

     Answered Sep 22, 2015 · 2 votes

     Unable to find any Matplotlib or Pandas examples to do the above, so had a go at writing something myself. First attempt attached with dummy data and explanatory comments. I'm not a professional programmer, so apologies for the inelegant style (I'm always grateful for the opportunity to learn any tips & constructive feedback from the experts on this forum, and to share my code for anyone to improve); but this example works, and hopefully it, or parts of it, will be useful to someone somewhere...

     import numpy as npimport matplotlib.pyplot as plt #--------------------------------def points(x,y,n): # Plot n points symmetrically aligned about axes dx=0.03 # define distance between individual dots m = 1-(n%2) # ensure symmetrical alignment for odd or even number of dots while(m<n): plt.scatter(x+(dx*m),y,color = 'k', marker = 'o', s=50, zorder=1) plt.scatter(x-(dx*m),y,color = 'k', marker = 'o', s=50, zorder=1) m+=2 return #-------------------------------- def histogram(b): # count number of data points in each bin for col in range(0,2): count = np.unique(b[:,col], return_counts=True) for n in range(0,np.size(count[col])): points(col,count[0][n], count[1][n]) return#------------------------------- def partition(a,bins): # partition continuous data into equal sized bins for plotting lo = np.min(a) hi = np.max(a) rng = hi-lo step = rng/float(bins-1)- for col in range (0,2): for row in range (0,int(np.size(a,axis=0))): for n in range (0,bins): if (a[row,col] <= (lo + (step / 2) + n * step)): b[row,col] = (lo + (n * step)) break return(b) #-------------------------------- def lines(b): # draw 'before' and 'after' lines between paired data points + median line for row in range (0,int(np.size(a,axis=0))): plt.plot([0,1],[b[row,0], b[row,1]], c='k',zorder=0, lw=1, alpha=0.3) plt.plot ([0,1],[np.median(b[:,0]),np.median(b[:,1])],c='r',zorder=2, lw=2, alpha=1) return#================================ # MAIN # Dummy paired continuous data (...or import from spreadsheet as a numpy array)\; a = np.array([ [1.62,1.53,1.42,1.39,1.11,1.20,0.99,0.88,0.60,0.65,0.52,0.49,0.43,0.41,0.31], # before[0.8,0.7,0.52,0.61,0.44,0.43,0.49,0.33,0.44,0.39,0.20,0.29,0.37,0.19,0.00] ]) # after-bins = 10 # choose total number of bins to categorise data intoax=plt.axes()-a = a.transpose()b=a # make a copy of the input data matrix to write categorised data tob = partition(a,bins) # partition continuous data into binslines(b) # draw lines between mid points of each bin and draw median linehistogram(b) # draw histograms centered at mid points of each bin-# Make general tweaks to plot appearance here:plt.xticks([0,1], ['OUT', 'IN'], fontsize=14)plt.ylabel('stimulation threshold (mA)',fontsize=13)plt.text(0.8,1.3,'All patients',fontsize=13)ax.patch.set_facecolor('white') # set background colour for plot area (default = white)ax.spines['top'].set_visible(False) # remove default upper axisax.spines['right'].set_visible(False) # remove default right axisplt.tick_params(axis='both',which='both',direction = 'out',top='off', right = 'off',labeltop='off') # remove tick marks from top & right axesplt.xlim(-0.6,1.6)plt.ylim(0,1.7)-plt.show() -

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   • Theorem numbering in LaTeX

     Top Answer

     Answered Jun 19, 2015 · 16 votes

     Putting the following code in the preamble seems to have the desired effect:

     \usepackage{amsthm}\newtheorem{thm}{Theorem}[subsection]\renewcommand{\thethm}{\arabic{subsection}.\arabic{thm}}-

     I don't understand why you want this particular theorem numbering system, but the code does what you want:

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   • Theorem and Proof Environment in Beamer

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     Answered Dec 13, 2022 · 3 votes

     Quarto very unhelpfully loads tons of unnecessary packages like amsthm, which beamer automatically loads, and then blindly tries to define a new theorem environment, even though beamer already has defined it. At the very minimum it should check if an environment is already defined before doing this.

     You can work around the problem with the notheorems class options.

     ---title: "Untitled"format: beamereditor: sourceclassoption: "notheorems"---## QuartoQuarto enables you to weave together content and executable code into a finished presentation. To learn more about Quarto presentations see <https://quarto.org/docs/presentations/>.::: {#thm-test}## Just a quick test:::

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2. www.geeksforgeeks.org › parallel-axis-theoremParallel Axis Theorem: Definition, History, Formula, Proof ...

   www.geeksforgeeks.org › parallel-axis-theorem

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   May 23, 2024 · Parallel axis theorem states that the moment of inertia of a rigid body about any axis parallel to its centroidal axis is equal to the sum of the body’s moment of inertia about its centroidal axis and the product of its mass and the square of the distance between the two axes.

3. Videos

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     Parallel Axis Theorem & Moment of Inertia - Physics Practice Problems

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     Statics: Lesson 68 - Parallel Axis Theorem, Area Moment of Inertia

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4. en.wikipedia.org › wiki › Moment_of_inertiaMoment of inertia - Wikipedia

   en.wikipedia.org › wiki › Moment_of_inertia

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   2 days ago · The moment of inertia of a thin disc of constant thickness , radius , and density about an axis through its center and perpendicular to its face (parallel to its axis of rotational symmetry) is determined by integration.

5. idealcalculators.com › parallel-axis-theoremParallel Axis Theorem Calculator - idealcalculators.com

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   May 14, 2024 · The Parallel Axis Theorem provides a method to compute the moment of inertia 𝐼I of an object about an axis parallel to a known axis that passes through its center of mass.

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7. www.vedantu.com › question-answer › state-and-deriveState and derive theorem for parallel and perpendicular axes.

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   Jun 3, 2024 · The parallel axis theorem states that, the moment of inertia of a body about any axis is equal to the moment of inertia about parallel axis through its center of mass plus the product of the mass of the body and the square of the perpendicular distance between the two parallel axes.

8. www.vedantu.com › jee-main › physics-perpendicularPerpendicular Axis Theorem – Explanation, Prove, Derivation ...

   www.vedantu.com › jee-main › physics-perpendicular

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   5 days ago · Perpendicular axis theorem statement - The perpendicular axis theorem states that the moment of inertia, for any axis which is perpendicular to the plane, is equal to the sum of any two perpendicular axes of the body which intersects with the first axis.

9. testbook.com › maths › perpendicular-axis-theoremUnderstanding Perpendicular Axis Theorem - Testbook

   testbook.com › maths › perpendicular-axis-theorem

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   May 22, 2024 · Explore the concept of Perpendicular Axis Theorem, its proof, applications, and difference with Parallel Axis Theorem. Also, get answers to frequently asked questions.

10. www.geeksforgeeks.org › moment-of-inertiaMoment of Inertia - GeeksforGeeks

    www.geeksforgeeks.org › moment-of-inertia

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    May 23, 2024 · According to Parallel Axis theorem, the moment of inertia of a body about a given axis is the sum of the moment of inertia about an axis passing through the center of mass of that body and the product of the square of the mass of the body and the perpendicular distance between the two axes.