Problem $3($HW $+2$ p$.)$

Wheel of radius R spins with constant angular velocity $$ as shown in the figure. What is the position, velocity, acceleration and the speed of the material point P at the instance $=45.$ Present your answer in $(1)$ cylindrical coordinate system and $(2)$ in the cartesian coordinate system shown in the figure.

Partial Answer:

Cylindrical frame: $,v=R{u}_{}$

Cartesian frame: $,v=\frac{R}{\sqrt[\phantom{2}]{2}}(-i+j)$

Problem $4$: $$

Particle P travels along a helix so that its position vector in the fixed xyz$-$frame is given by $r(t)=r(t)cos(t)i+r(t)sin(t)j+z(t)k,$ where $r(t)=R,(t)=t$ and $z(t)=$ and $$ are all constants$).$ Derive the expression for the position $r,$ velocity $v=r^{}$ and acceleration $a=r^{}$ of $P$ in the cylindrical coordinate system.

Answer:

$r=R{u}_r+v_{z}t{u}_z$

$v=R{u}_{}+v_z{u}_z$

$a=-R{}^2{u}_r$