Q. 63 Prove that the magnitude of the ... [FREE SOLUTION]

Chapter 11: Q. 63 (page 890)

Prove that the magnitude of the acceleration of a particle moving along a curve, C, with a constant speed is proportional to the curvature of C.

Short Answer

Expert verified

It is proved thatthe magnitude of the acceleration of a particle moving along a curve,C, with a constant speed is proportional to the curvature ofC.

Step by step solution

Step 1. Given Information.

The formula for the Curvature of a Space Curve is $魏 = \frac{||r' (t) 脳 r'' (t)||}{{||r' (t)||}^{3}} .$

Step 2. Prove.

To prove that the magnitude of the acceleration of a particle moving along a curve, C, with a constant speed is proportional to the curvature of C,we will use the formula of curvature 魏 of C at a point on the curve.

So,

$魏 = \frac{||r' (t) 脳 r'' (t)||}{{||r' (t)||}^{3}} 魏 = \frac{||r' (t)|| ||r'' (t)|| \sin 胃}{{||r' (t)||}^{3}} 魏 = \frac{||r' (t)|| ||r'' (t)|| \sin \frac{蟺}{2}}{{||r' (t)||}^{3}} [speed is constant] 魏 = \frac{||r' (t)|| ||r'' (t)||}{{||r' (t)||}^{3}} [\sin \frac{蟺}{2} = 1] 魏 = \frac{||r'' (t)||}{{||r' (t)||}^{2}}$

Now, the denominator of kis the square of the speed, which we presumed as constant.

Thus,

$魏鈭� ||r'' (t)|| .$

Hence proved.

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91影视

Prove that the magnitude of the acceleration of a particle moving along a curve, C, with a constant speed is proportional to the curvature of C.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Prove.

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