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- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/11%3A_Simple_and_Damped_Oscillatory_Motion/11.05%3A_Damped_Oscillatory_Motion/11.5iii%3A_Critical__damping-_(_gamma_%3D_2omega_0)In order that the coil and the pointer should move to the equilibrium position in the fastest possible time without oscillating, the system should be critically damped - which means that the rotationa...In order that the coil and the pointer should move to the equilibrium position in the fastest possible time without oscillating, the system should be critically damped - which means that the rotational inertia and the electrical resistance of the little aluminium former has to be carefully designed to achieve this.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/16%3A_Hydrostatics/16.05%3A_Pressure_on_a_Vertical_SurfaceThe total force on a submerged vertical or inclined plane surface is equal to the area of the surface times the depth of the centroid.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/02%3A_Moments_of_Inertia/2.14%3A_Eigenvectors_and_EigenvaluesThe vector \( \left(\begin{array}{c}1\\ 1\end{array}\right) \) is a very special one, and it is called an eigenvector of the matrix, and the multiplier 3 is called the corresponding eigenvalue. In sho...The vector \( \left(\begin{array}{c}1\\ 1\end{array}\right) \) is a very special one, and it is called an eigenvector of the matrix, and the multiplier 3 is called the corresponding eigenvalue. In short, given a square matrix A, if you can find a vector a such that Aa = \( \lambda \) a, where \( \lambda \) is merely a scalar multiplier that does not change the direction of the vector a, then a is an eigenvector and \( \lambda \) is the corresponding eigenvalue.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/07%3A_ProjectilesThumbnail: A photo of a Smith & Wesson firing, taken with an air-gap flash. The photo was taken in a darkened room, with camera's shutter open and the flash was triggered by the sound of the shot usin...Thumbnail: A photo of a Smith & Wesson firing, taken with an air-gap flash. The photo was taken in a darkened room, with camera's shutter open and the flash was triggered by the sound of the shot using a microphone. (CC BY-SA 3.0; Niels Noordhoek).
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/10%3A_Rocket_Motion/10.02%3A_An_IntegralSo that we don't get bogged down later with an integral that is going to crop up, see if you can do the following integration: \[ \int\ln(a-bt)dt \nonumber \] You should get \( -t-\frac{1}{b}(a-bt)\ln...So that we don't get bogged down later with an integral that is going to crop up, see if you can do the following integration: \[ \int\ln(a-bt)dt \nonumber \] You should get \( -t-\frac{1}{b}(a-bt)\ln(a-bt)+\) constant.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/06%3A_Motion_in_a_Resisting_MediumIn studying the motion of a body in a resisting medium, we assume that the resistive force on a body, and hence its deceleration, is some function of its speed. Such resistive forces are not generally...In studying the motion of a body in a resisting medium, we assume that the resistive force on a body, and hence its deceleration, is some function of its speed. Such resistive forces are not generally conservative, and kinetic energy is usually dissipated as heat.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/18%3A_The_CatenaryIf a flexible chain or rope is loosely hung between two fixed points, it hangs in a curve that looks a little like a parabola, but in fact is not quite a parabola; it is a curve called a catenary, whi...If a flexible chain or rope is loosely hung between two fixed points, it hangs in a curve that looks a little like a parabola, but in fact is not quite a parabola; it is a curve called a catenary, which is a word derived from the Latin catena, a chain.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/10%3A_Rocket_MotionThumbnail: A Trident II missile launched from sea. Image use with permission (Public Domain; Department of Defense).
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/15%3A_Special_Relativity/15.11%3A_The_Twins_ParadoxDuring the late 1950s and early 1960s there was great controversy over a problem known as the “Twins Paradox”. The controversy was not confined to within scientific circles, but was argued, by scienti...During the late 1950s and early 1960s there was great controversy over a problem known as the “Twins Paradox”. The controversy was not confined to within scientific circles, but was argued, by scientists and others, in the newspapers, magazines and many serious journals. . The paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/22%3A_Dimensions/22.03%3A_Checking_EquationsIf the equation does not balance dimensionally, you know immediately that you have made a mistake, and the dimensional imbalance may even give you a hint as to what the mistake is. Suppose that you ha...If the equation does not balance dimensionally, you know immediately that you have made a mistake, and the dimensional imbalance may even give you a hint as to what the mistake is. Suppose that you have deduced (or have read in a book) that the period of oscillations of a torsion pendulum is \( P = 2 \pi \sqrt{\frac{I}{C}} \), where \(I\) is the rotational inertia and \(c\) is the torsion constant.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/01%3A_Centers_of_Mass/1.04%3A_Plane_Curves\[ \delta s = \sqrt{( \delta r)^{2} + (r \delta \theta )^{2}} = \sqrt{\left( \dfrac{dr}{d \theta }\right)^{2} + r^{2}} \, \delta \theta = \sqrt{1+\left(r \dfrac{d \theta }{dr}\right)^{2}} \,\delta r. ...\[ \delta s = \sqrt{( \delta r)^{2} + (r \delta \theta )^{2}} = \sqrt{\left( \dfrac{dr}{d \theta }\right)^{2} + r^{2}} \, \delta \theta = \sqrt{1+\left(r \dfrac{d \theta }{dr}\right)^{2}} \,\delta r. \label{eq:1.4.5} \] The equation in polar coordinates is simply \( r = a \), and the integration limits are \( \theta = \dfrac{- \pi}{2} \) to \( \theta = \dfrac{+ \pi}{2} \) and the length is \( \pi a \).
