Problem $1$

Two solid spherical masses $m_1=2\mathrm{.}51kg$ and $m_3=1\mathrm{.}48kg$ with radii $r_1=0\mathrm{.}21$ and $r_3=0\mathrm{.}14,$ are attached to the end of a massive rigid rod of mass $m_2=1\mathrm{.}00kg,$ as shown in the below figure. The rod is attached to a massive cylinder of mass $m_4=3\mathrm{.}2kg$ and radius $r_4=0\mathrm{.}14m,$ and is free to rotate in a vertical plane about a frictionless axle perpendicular to the rod through point $P$ a distance $l_1=0\mathrm{.}83m$ and $l=0\mathrm{.}52m.$ The bar is released from rest in the horizontal position at $t=0.$

a$)$ Find the moment of inertia of the rod with all three masses.

b$)$ Find the torque acting on the system at $t=0.$

c$)$ Find the angular acceleration of of the system at $t=0.$

d$)$ Find the maximum kinetic energy of the system during the swing.

e$)$ Find the maximum angular velocity of the system.

Variable Type Data$/$Info

I$1$

float

$1\mathrm{.}7734153999999998$Problem $1$

Two solid spherical masses $m_1=2\mathrm{.}51kg$ and $m_3=1\mathrm{.}48kg$ with radii $r_1=0\mathrm{.}21$ and $r_3=0\mathrm{.}14,$ are attached to the end of a massive rigid rod of mass $m_2=1\mathrm{.}00kg,$ as shown in the below figure. The rod is attached to a massive cylinder of mass $m_4=3\mathrm{.}2kg$ and radius $r_4=0\mathrm{.}14m,$ and is free to rotate in a vertical plane about a frictionless axle perpendicular to the rod through point $P$ a distance $l_1=0\mathrm{.}83m$ and $l=0\mathrm{.}52m.$ The bar is released from rest in the horizontal position at $t=0.$

a$)$ Find the moment of inertia of the rod with all three masses.

b$)$ Find the torque acting on the system at $t=0.$

c$)$ Find the angular acceleration of of the system at $t=0.$

d$)$ Find the maximum kinetic energy of the system during the swing.

e$)$ Find the maximum angular velocity of the system.

Variable Type Data$/$Info

I$1$

float

$1\mathrm{.}7734153999999998$

Variable Type Data$/$Info

$\backslash$table$[[$I$1,$float,$1\mathrm{.}7734153999999998],[$I$2,$float,$0\mathrm{.}1759],[$I$3,$float,$0\mathrm{.}41179520000000003],[$I$4,$float,$0\mathrm{.}031360000000000006],[$MoI$,$float,$2\mathrm{.}3924706],[$PE$,$float,$14\mathrm{.}393259999999994],[$alpha$,$float,$6\mathrm{.}0160655683710385],[$d$,$float,$0\mathrm{.}15499999999999997],[$g$,$float,$9\mathrm{.}8],[$l$1,$float,$0\mathrm{.}83],[$l$2,$float,$0\mathrm{.}52],[$m$1,$float,$2\mathrm{.}51],[$m$2,$int,$1],[$m$3,$float,$1\mathrm{.}48],[$m$4,$float,$3\mathrm{.}2],[$r$1,$float,$0\mathrm{.}21],[$r$3,$float,$0\mathrm{.}14],[$r$4,$float,$0\mathrm{.}14],[$tau$,$float,$14\mathrm{.}393259999999998]]$

Variable Type Data$/$Info

$\backslash$table$[[$I$1,$float,$1\mathrm{.}7734153999999998],[$I$2,$float,$0\mathrm{.}1759],[$I$3,$float,$0\mathrm{.}41179520000000003],[$I$4,$float,$0\mathrm{.}031360000000000006],[$MoI$,$float,$2\mathrm{.}3924706],[$PE$,$float,$14\mathrm{.}393259999999994],[$alpha$,$float,$6\mathrm{.}0160655683710385],[$d$,$float,$0\mathrm{.}15499999999999997],[$g$,$float,$9\mathrm{.}8],[$l$1,$float,$0\mathrm{.}83],[$l$2,$float,$0\mathrm{.}52],[$m$1,$float,$2\mathrm{.}51],[$m$2,$int,$1],[$m$3,$float,$1\mathrm{.}48],[$m$4,$float,$3\mathrm{.}2],[$r$1,$float,$0\mathrm{.}21],[$r$3,$float,$0\mathrm{.}14],[$r$4,$float,$0\mathrm{.}14],[$tau$,$float,$14\mathrm{.}393259999999998]]$

Problem $1$

Two solid spherical masses $m_1=2\mathrm{.}51kg$ and $m_3=1\mathrm{.}48kg$ with radii $r_1=0\mathrm{.}21$ and $r_3=0\mathrm{.}14,$ are attached to the end of a massive rigid rod of mass $m_2=1\mathrm{.}00kg,$ as shown in the below figure. The rod is attached to a massive cylinder of mass $m_4=3\mathrm{.}2kg$ and radius $r_4=0\mathrm{.}14m,$ and is free to rotate in a vertical plane about a frictionless axle perpendicular to the rod through point $P$ a distance $l_1=0\mathrm{.}83m$ and $l=0\mathrm{.}52m.$ The bar is released from rest in the horizontal position at $t=0.$

a$)$ Find the moment of inertia of the rod with all three masses.

b$)$ Find the torque acting on the system at $t=0.$

c$)$ Find the angular acceleration of of the system at $t=0.$

d$)$ Find the maximum kinetic energy of the system during the swing.

e$)$ Find the maximum angular velocity of the system.

Variable Type Dat$\frac{a}{\text{I}}$nfo

$I1$ float $1\mathrm{.}7734153999999998$