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Wewaii [24]
3 years ago
5

A particle moves so that its position vector with respect to the origin of a reference frame Oxyz is r(t)=bcos w t+bsin w t+vt k

,where i,j and k are unit vectors parallel to the co-ordinates axes Ox,Oy and b&V are positive constants.
(i)Find the velocity and speed of the particle
(ii)Describe the path moved by the particle
(iii)Find the acceleration of the particle

​
Mathematics
1 answer:
sweet-ann [11.9K]3 years ago
3 0

(i) Velocity is the rate of change of position, so if

<em>r</em><em>(t)</em> = <em>b</em> cos(<em>ω t </em>) <em>i</em> + <em>b</em> sin(<em>ω t </em>) <em>j</em> + <em>v</em> <em>t</em> <em>k</em>

then

<em>v</em><em>(t)</em> = d<em>r</em>/d<em>t</em>

<em>v</em><em>(t)</em> = -<em>b</em> <em>ω </em>sin(<em>ω t</em> ) <em>i</em> + <em>b</em> <em>ω</em> cos(<em>ω</em> <em>t</em> ) <em>j</em> + <em>v</em> <em>k</em>

The speed of the particle is the magnitude of the velocity, given by

|| <em>v</em><em>(t)</em> || = √[(-<em>b</em> <em>ω </em>sin(<em>ω t</em> ))² + (<em>b</em> <em>ω</em> cos(<em>ω</em> <em>t</em> ))² + <em>v</em> ²]

… = √[<em>b </em>²<em>ω </em>² + <em>v</em> ²]

(ii) The path is a helix. Suppose you zero out the <em>k</em> component. Then the path is a circle of radius <em>b</em>, and the value of <em>ω</em> determines how quickly a particle on the path traverses the circle. Now if you reintroduce the <em>k</em> component, the value of <em>v</em> will determine how far from the plane <em>z</em> = 0 the particle moves in a helical path as <em>t</em> varies.

(iii) Acceleration is the rate of change of velocity, so

<em>a</em><em>(t)</em> = d<em>v</em>/d<em>t</em>

<em>a</em><em>(t)</em> = -<em>b</em> <em>ω </em>²<em> </em>cos(<em>ω t</em> ) <em>i</em> - <em>b</em> <em>ω</em> ² sin(<em>ω</em> <em>t</em> ) <em>j</em>

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