Cycloid. A particle moves in the xy-plane. Its coordinates are given as function

Question

Cycloid. A particle moves in the xy-plane. Its coordinates are given as functions of time by:
x(t) = R (wt sin wt)
y(t) = R (1 cos wt)
where R and w are constants.

Determine the velocity x component of the particle at any time .
=


Part B
Determine the velocity -component of the particle at any time .
=


Part C
Determine the acceleration -component of the particle at any time .
=

Part D
Determine the acceleration -component of the particle at any time .

=


Part E
At which times is the particle momentarily at rest?
=


Part F
What is the -coordinate of the particle at these times?


Part G
What is the -coordinate of the particle at these times?
=

Part H
What is the magnitude of the acceleration at these times?

Explanation / Answer

b) Vx=R(w-wcoswt) Vy=R(wsinwt) Ax=R(w^2sinwt) Ay=R(w^2coswt) c) At rest for Vx=0 and Vy=0 Vx=0=R(w-wcoswt) ie. when coswt =1; wt =0, 2Pi, 4Pi... Vy=0=R(wsinwt) i.e. when sinwt =0; wt = Pi/2, 3Pi/2, 5Pi/2... Substitute these wt values into the x(t), y(t) equations to get the coordinates. d) The magnitude of the acceleration is Rw^2 and does not depend on time. Compared to uniform circular motion it is related by the speed of the vehicle if the vehicle speed were deducted then the motion would be circular.

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