A hanging mass m$=82$kg

is released from rest and accelerates downward. The winding drum A

of mass ma$=65$kg

with an effective diameter of $\backslash$Phi a$=440$mm

and a radius of inertia ka$=350$mm

meshes with gear B

$.$

$>>>$ Please note: Gear B

has a moment of inertia of $0\mathrm{.}9$ kgm$2$

$.$

Assume $\backslash$omega b$=5\backslash$omega a

$,$ with subscript a

related to the winding drum and b

to gear B

$.$

Analyse the frictionless system and calculate:

The moment of Inertia of the drum Ia$=$mak$2$a$=$

Answer for part $1$

$7\mathrm{.}9625$

One possible correct answer is: $7\mathrm{.}9625$

kgm$2$

The linear acceleration a$=$

Answer for part $2$

$3\mathrm{.}2632$

One possible correct answer is: $1\mathrm{.}1307713621037$

m$/$s$/$s

The angular accelerations for the drum $\backslash$alpha a$=$

Answer for part $3$

$14\mathrm{.}833$

One possible correct answer is: $5\mathrm{.}1398698277439$

rad$/$s$/$s

as well as the gear $\backslash$alpha b$=$

Answer for part $4$

$74\mathrm{.}159$

One possible correct answer is: $25\mathrm{.}69934913872$

rad$/$s$/$s

$(*)$

The tension in the cord F$1=$

Answer for part $5$

$267\mathrm{.}566$

One possible correct answer is: $711\mathrm{.}6967483075$

N

$.$

Assume an observation period of $2$ s

from the instant the system is released from rest and calculate the change of linear speed $\backslash$Delta v$=$

Answer for part $6$

$6\mathrm{.}526$

One possible correct answer is: $2\mathrm{.}2615427242073$

m$/$s

and linear distance the hung mass moves $\backslash$Delta h$=$

Answer for part $7$

$6\mathrm{.}526$

One possible correct answer is: $2\mathrm{.}2615427242073$

m

$(*)$

During the $2$ s

observation, the following changes of energy with the correct sign can be computer:

Ek$=$

Answer for part $8$

$1746\mathrm{.}136$

One possible correct answer is: $2102\mathrm{.}23847006$

J

$($Kinetic energy$)$ and

Ep$=$

Answer for part $9$

$5249\mathrm{.}645$

One possible correct answer is: $-1819\mathrm{.}2301982069$

J

$($Potential Energy$)$

The perfect, frictionless system described above is replaced with a realistic one with a gear efficiency of $92\%$

$.$

Analyse the impact of the efficiency of the dynamics of the system and calculate:

The acceleration of the system under new conditions, a$2=$

Answer for part $10$

One possible correct answer is: $1\mathrm{.}0699713843583$

m$/$s$/$s

$,$

the tension T$=$

Answer for part $11$

One possible correct answer is: $716\mathrm{.}68234648262$

N

and

the number of revolutions of the drum during the same period of observation of $2$ sec

: N$=$

Answer for part $12$

One possible correct answer is: $1\mathrm{.}5481021344318$

rev