Chapter 11: Q. 17 (page 898)

Find
(a) the displacement vectors from r(a) tor(b),
(b) the magnitude of the displacement vector, and
(c) the distance travelled by a particle on the curve from a to b.
r(t) = α sinβt,α cosβt,γt, a = 0, b = 1

Short Answer

Expert verified

$(a) The displacement vector from r (a) t o r (b) i s <α \sin β, α (\cos β - 1), γ> (b) Hence the magnitude of the displacement vector is = \sqrt{2 α^{2} (1 - \cos β) + γ^{2}} (c) The distance travelled from a to b is = \sqrt{α^{2} β^{2} + γ^{2}}$

Step by step solution

Part (a) Step 1. Given Information

The given function is r(t) = α sinβt,α cosβt,γt, a = 0, b = 1

Part (a) Step 2. Finding the displacement vector

$r (t) = α \sin β t, α \cos β t, γ t, a = 0, b = 1 The displacement vector from r (a) t o r (b) = r (b) - r a (a) = r (1) - r (0) = <α \sin β, α \cos β, γ> - <α \sin β 0, α \cos β 0, γ (0),> = <α \sin β, α \cos β, γ> - <0, α, 0> = <α \sin β, α (\cos β - 1), γ> The displacement vector from r (a) t o r (b) i s <α \sin β, α (\cos β - 1), γ>$

Part (b) Step 1. Finding the magnitude of the displacement vector,

$The magnitude of displacement vector ||<α \sin β, α (\cos β - 1), γ>|| = \sqrt{α^{2} \sin^{2} β, α^{2} {(\cos β - 1)}^{2}, γ^{2}} = \sqrt{2 α^{2} (1 - \cos β) + γ^{2}} Hence the magnitude of the displacement vector is = \sqrt{2 α^{2} (1 - \cos β) + γ^{2}}$

Part (c) Step 1. Finding the displacement vector

$r (t) = ⟨ α \sin π t, α \cos β t, γ t ⟩ ||r^{'} (t)|| = \sqrt{α^{2} β^{2} \cos^{2} β t + α^{2} β^{2} \sin^{2} β t + γ^{2}} = \sqrt{α^{2} β^{2} (\cos^{2} β t + \sin^{2} β t) + γ^{2}} = \sqrt{α^{2} β^{2} + γ^{2}} The distance travelled from a to b is = \int_{0}^{1} \sqrt{α^{2} β^{2} + γ^{2}} d t = (\sqrt{α^{2} β^{2} + γ^{2}}) (1 - 0) substitute the limits for t = \sqrt{α^{2} β^{2} + γ^{2}} The distance travelled from a to b is = \sqrt{α^{2} β^{2} + γ^{2}}$

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Find
(a) the displacement vectors from r(a) tor(b),
(b) the magnitude of the displacement vector, and
(c) the distance travelled by a particle on the curve from a to b.
r(t) = α sinβt,α cosβt,γt, a = 0, b = 1

Short Answer

Step by step solution

Part (a) Step 1. Given Information

Part (a) Step 2. Finding the displacement vector

Part (b) Step 1. Finding the magnitude of the displacement vector,

Part (c) Step 1. Finding the displacement vector

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Find(a) the displacement vectors from r(a) tor(b),(b) the magnitude of the displacement vector, and(c) the distance travelled by a particle on the curve from a to b.r(t) = α sinβt,α cosβt,γt, a = 0, b = 1

Short Answer

Step by step solution

Part (a) Step 1. Given Information

Part (a) Step 2. Finding the displacement vector

Part (b) Step 1. Finding the magnitude of the displacement vector,

Part (c) Step 1. Finding the displacement vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Pure Maths

Decision Maths

Calculus

Study anywhere. Anytime. Across all devices.

Find
(a) the displacement vectors from r(a) tor(b),
(b) the magnitude of the displacement vector, and
(c) the distance travelled by a particle on the curve from a to b.
r(t) = α sinβt,α cosβt,γt, a = 0, b = 1