Problem Statement

An $$electric motor is $$attached to $$the shaft of $$a winch, which is $$used to $$lift a crate as $$shown below. The motor provides a

torque M$$to $$the winch, which is $$connected to $$the crate through a single pulley. The system starts from rest. The following

parameters are known:

Wp$=8$lb $$weight of $$pulley

Ww$=75$lb $$weight of $$winch

Wc$=495$lb $$weight of $$crate

rw$=15$in $$radius of $$winch drum $($around $$which cable wraps$)$

rp$=6$in $$radius of $$pulley

kw$=6$in $$radius of $$gyration of $$pulley

kp$=1$in $$radius of $$gyration of $$winch

The torque applied by $$the motor to $$the winch is:

M$=-0\mathrm{.}1$$2+315$

where is $$the winch speed in$$rads $$and the torque M$$is$$in$$ft$-$lbf$.$

Assignment

Use Conservation of $$Energy to $$develop a differential equation for the speed of $$the crate $($or $$the rotational speed of $$the

winch$)$as $$a function of $$time$.$

Solve the resulting differential equation to $$determine V$($t$)$; show your solution both as$$an $$explicit analytical equation

and as $$a plot for t$$$0$s$.$

Validation

$($a$)$Use Conservation of $$Momentum to $$develop the governing differential equation, and show that it$$is $$the same as $$that

obtained from Conservation of $$Energy$.[$Hint: while it $$may not be $$intuitive$,$ since the pulley has an $$angular

acceleration, the tension in $$the two halves of $$the cable connecting it$$to $$the winch are not the same.$]$

$($b$)$Solve the differential equation numerically using Euler's Method and compare to $$the analytical solution.

$($c$)$Solve the steady$-$state problem to $$determine the steady$-$state speed of $$the crate, and compare to $$the solution of $$the

transient problem $($the $$original problem$)$when t$$$.$