User:JohnBlackburne/test

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 * (とある科学の)
 * ( とある科学の 超電磁砲【レールガン】 )

$$v_x^2 + v_y^2 = s^2$$

$$\begin{align}{v_x}t &= r\\{v_y}t - \tfrac{gt^2}{2} &= 0\end{align}$$

$$t^2 = 2\left(\frac{s^2 + hg + \sqrt((s^2 + hg)^2 - g(r^2 + h^2))}{g^2}\right)$$

$$(x_0 - a)^2 + (y_0 - b)^2 < r^2\,$$

$$(a_0 - a_1)^2 + (b_0 - b_1)^2 < (r_0 + r_1)^2\,$$

$$(a_0 - a_1)^2 + (b_0 - b_1)^2 < (r_0 - r_1)^2\,$$


 * $$\theta=\arccos \left( \frac {\mathbf{a}\cdot\mathbf{b}} {\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|}\right)$$

$$d = \mathbf{n} \cdot (\mathbf{x} - \mathbf{a})\,$$

$$ \mathbf{x} = \mathbf{x} + 2d\mathbf{n}$$

$$s = -\mathbf{n} \cdot \mathbf{v}\,$$

$$\mathbf{v} = \mathbf{v} + s(1 + e)\mathbf{n}$$

$$t = \tan \frac{\theta}{2} $$

$$\sin \theta = \frac{2t}{1 + t^2}, \cos \theta = \frac{1-t^2}{1 + t^2}, \tan \theta = \frac{2t}{1 - t^2}$$

Normal reaction in a banked turn
For a vehicle is stationary on banked turn, or more generally on an inclined plane, the normal force is


 * $$mg\cos \theta,\,$$

where θ is the bank angle or the angle of elevation of the surface. The vehicle also requires a frictional force acting up the slope, with magnitude


 * $$mg\sin \theta,\,$$

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But once the car starts to move on a banked turn, it 'collides' with the turn itself, the turn feels this force, and returns it in the normal, causing the car to move circularly.

The Normal is thus greater than simply its gravitational component.


 * $$N = mg\cos \theta + {mv^2\over r} \sin \theta$$

If the Normal were simply the component to gravity, one could not say mg = Ncosθ = mgcos2θ when the vertical acceleration is 0.

Old style

New style

end

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🀙🀚🀛🀜🀝🀞🀟🀠🀡

Hello


 * 1) 010203

a
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$$ \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ A rotation through 90° in the xy-plane

$$ \begin{pmatrix} \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ 0 & 1 & 0 \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{pmatrix}$$ A rotation through 45° in the xz-plane

$$ \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$$ An even permutation matrix which permutes the vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1), and is an 120° rotation in the plane parallel to the one through these three points.

The following translation is taken from E.G. Millar 1923 The Lindisfarne Gospels London: British Museum and he got it from F. E. Harmer 1914 Select English Historical Documents of the Ninth and Tenth Centuries, No. xxii, Cambridge.


 * Eadfrith, Bishop of Lindisfarne, wrote this book, at the first, in honour of God and St Cuthbert and all the saints in common who are on the island. And Aethelwald, Bishop of Lindisfarne, bound it on the outside and covered (?) it, as he was well able to do. And Billfrith, the anchorite, wrought the ornaments upon the outside and adorned it, this unalloyed metal gilded over, with gold and gems and also with silver (?). And Aldred, an unworthy and most miserable priest, with the help of God and St Cuthbert, wrote an English gloss above, and obtained for himself a home (?) with the three parts; [he glossed] Matthew's part for God and St Cuthbert, Mark's part for the bishop, and Luke's for the community, paying, in addition, eight 'ores' of silver for his admission(?). And St John's part [he glossed] for himself, namely for the good of his soul, and has offered to God and St Cuthbert four 'ores' of silver besides; that he may receive admission in Heaven through God's mercy, and have happiness and peace upon earth, promotion and honour, wisdom and prudence, through the merits of St Cuthbert. Eadfrith, Aethelwald, Billfrith, Aldred, made or adorned this Gospel Book for God and Cuthbert.

A slightly variant translation of the first part of the text is provided by J. Backhouse 1981 The Lindisfarne Gospels Oxford: Phaidon.


 * Eadfrith, Bishop of the Lindisfarne Church, originally wrote this book, for God and for Saint Cuthbert and - jointly- for all the saints whose relics are on the island. And Ethelwald, Bishop of the Lindisfarne islanders, impressed it on the outside and covered it - as he well knew how to do. And Billfrith, the anchorite, forged the ornaments which are on it on the outside and adorned it with gold and with gems and also with gilded-over silver - pure metal. And Aldred, unworthy and most miserable priest, glossed it in English between the lines with the help of God and Saint Cuthbert ....

This just goes to show that translation is a tricky art, especially with an unusual text, and involves a great deal of interpretation.

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