Question

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

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Answer 1

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

r(t) = b cos(ω t ) i + b sin(ω t ) j + v t k

then

v(t) = dr/dt

v(t) = -b ω sin(ω t ) i + b ω cos(ω t ) j + v k

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

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

… = √[b ²ω ² + v ²]

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

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

a(t) = dv/dt

a(t) = -b ω ² cos(ω t ) i - b ω ² sin(ω t ) j