Homework H$\mathrm{.}4.$B

Given: Particle P $($of mass $m)$ is attached to an inextensible cable, with the opposite end of the

cable being attached to a fixed pin O$.$ P is also constrained to move along the rough surface of a

right circular cone, with P following a circular path of radius $r.$ The coefficient of kinetic friction

between P and the cone is known to be ${}_k.$ The angle of cable OP with respect to the vertical is

known to be $.$ At a particular instant in time, P is moving with a speed of $v.$

Find: For this instant in time,

$($a$)$ Determine the normal component of force of the cone acting on particle P $.$

$($b$)$ Determine the rate of change of speed of particle $P.$

TOP view of the path of $P$

Use the following parameters in your analysis: ${}_k=0\mathrm{.}3,m=10kg,=36\mathrm{.}87,r=0\mathrm{.}5m$ and $v$

$=1\mathrm{.}5\frac{m}{s}.$