Mechanical Principles - Rotating Systems

[louise]

A load of mass 250kg is required to be lifted by a means of a winding drum and cable. The mass will be initially at rest, accelerated uniformly upwards for 4 seconds and then decelerated uniformly for one second. such that a final height of 10.5 metres is gained.

The winding drum has a mass of 225kg, a diameter of 1.5m and a radius of gyration of 320mm. The bearings of the drum have a constant frictional torque of 5Nm.

Calculate:
a) The maximum velocity reached by the mass.
b) The maximum angular velocity of the winding drum.
c) The total work done whilst accelerating the load upwards.
d) The input torque to the driving motor whilst accelerating the load.
e) The average power required whilst accelerating the load.
f) The maximum power required from the drive motor.

My solutions so far:
a) Using: s=ut+1/2at^2 (1/2 being half and ^2 being squared)
a=2(s-ut)/t^
=2(10.5-0*4)/4^2
= 1.31m/s^2
Using: v=u+at
=0+1.13*4
=5.24m/s^2
b) Using: θ=((ω1+ω2)t)/2 UNSURE ON THIS ON FOR DEFINITE!
=((0+5.24)4)/2
=10.48radians?
θ/t=Angular Velocity
10.48/2∏ (∏ being used as Pi)
=16.46revs

c) NEED HELP I HAVE THIS EQUATION BUT I NEED TO FIGURE OUT ALL THE LETTERS FIRST
WD=mg(h2-h1)+1/2m(v^2-u^2)+1/2I(ω2^2-ω1^2)+(Fr*θ)
I know mK^2 =I m being the mass and K being the radius of gyration?
An to find h2 I use S=((u+v)t)/2
d)Ang Power =WD/t?
f)??
g) Max Power = T*ω1??

PROBABLY ALL WRONG - PLEASE HELP, QUITE BAFFLED AT THE MOMENT!

 

[Simon Bridge]

Welcome to PF;
Your overall strategy appears to involve trying to find the right equations to use.
It is better to use your understanding of the physics.

(a) fair enough - you can also draw a v-t diagram.
(b) what is the relationship between angular velocity and linear velocity?
(c) ignore the formula - work = change in energy: where does the energy put into winding the drum go?
(d) W=Fd normally - what is the angular form?
(e&f) what is the definition of power?

note: if you know the linear equation, you can get the equivalent rotational one by substituting:
velocity -> angular velocity
acceleration -> angular acceleration
force -> torque
displacement -> angle

 

[louise]

Yes, I am looking for help with the equations and advice as to whether the methods I've attempted to use are correct.

a) I have already done a velocity time graph to represent the two in relation with each other. Wasn't too sure that my working for the calculations were correct.
b) They are the same, but because the winding drum is the rotating object you can't use the same formula. That's where i was confused.
c) Into the load? I have no idea.
d) I needed the average power not angular sorry! W=Fd? We've never being taught that. What does the lettering stand for?
e&f) WD/time = Pavg Torque*?=Pmax

 

[louise]

PS. I'm not a whizz at physics and the formulas involved with Mech Principles, its just one unit of many we have to do.

 

[Simon Bridge]

Yes, I am looking for help with the equations and advice as to whether the methods I've attempted to use are correct.

a) I have already done a velocity time graph to represent the two in relation with each other. Wasn't too sure that my working for the calculations were correct.

Oh good - well from the v-t graph you just do a bit of geometry to check your result via equations.

b) They are the same, but because the winding drum is the rotating object you can't use the same formula. That's where i was confused.

No ... linear velocity and angular velocity are not the same - they have a reationship.
Have you see v=rω?

c) Into the load? I have no idea.

Come on - what kinds of energy are there?
Is the load moving vertically? What kind of energy changes?
Is the load changing speed? What kind of energy changes?
Is there any friction?

d) I needed the average power not angular sorry! W=Fd? We've never being taught that. What does the lettering stand for?

It is the definition of Work.
W = work
F = force
d = displacement

 

[dean barry]

M = load mass = 250 kg
m = drum mass = 225 kg
k = radius of gyration = 0.32 metres

Im assuming the mass moment of inertia (i) = m * k ² = 225 * 0.32 ² = 23.04 kg-m²
----------------------------------------------------------------------------------------------------
There is a short cut that might help you cut out some work, you can give the load and drum a combined equivalent mass (EM) for use in this type of problem :

Give the load an example constant velocity, say 10 m/s and calculate the linear KE from :
KE linear = ½ * M * v ²

Calculate the rotation rate ω ( radians / second ) of the drum with the load at 10 m/s from v / r
Calculate the rotary KE of the drum from :
KE rotary = ½ * i * ω ²

Install in the following equation :
EM (in kgs) = ( 1 + ( KE rotary / KE linear ) ) * M

If you have a linear acceleration rate (a) for the load for instance, you can calculate the force (f) required to accelerate both the load and the drum from :
f = EM * a

im uncertain as to the performance you require under accelerating and braking, can you post the graph you have ?
thanks
dean

 

[dean barry]

Sorry louise, led you up the wrong path there, disregard my input.
dean