What Is the Effect of Torque on Pulley Systems with Friction?

[bluebear19]

Homework Statement

The two blocks are connected by a massless rope that passes over a pulley. The pulley is 17 cm in diameter and has a mass of 2.6 kg . As the pulley turns, friction at the axle exerts a torque of magnitude 0.51 Nm.

Homework Equations

http://answers.yahoo.com/question/index?qid=20081114102306AA0xfMk
i followed the advice given here but it doesn't seem to work

The Attempt at a Solution

i know how to do the problem without torque but I don't know how to add the torque in, I've been doing it for long time and haven't been getting the right answer
thanks in advance

 

[Hootenanny]

Whilst I haven't read through the entire response at yahoo, I did spot one obvisous error in your calculations

t = sqrt[2ah] = sqrt[2*4.12*1] = 2.87s

Are you sure that you should multiply by the acceleration?

 

[bluebear19]

the response on yahoo used that formula, i don't know what i should use, should it be. thanks for helping

 

[Hootenanny]

the response on yahoo used that formula, i don't know what i should use, should it be. thanks for helping

Indeed, the response from Yahoo does appear to make use of that formula. However, the formula contains a typo. The correct formula can be derived from the following kinematic equation

h = v_0t +\frac{1}{2}at^2

 

[bluebear19]

but isn't the Vo = 0 zero since they are dropped from rest so solving for t so that should give you t = sqrt[2h/a], is that right now?

 

[bluebear19]

but then how come in the yahoo answer, the time was 1.7s?? using this formula i only get that t = around0.7 s? can u tell me what i am doing wrong please thanks so much

 

[Hootenanny]

It would be much easier to point out your mistake if you could post your working.

 

[bluebear19]

i did in the yahoo link. i replied to the answer with my own workings

 

[bluebear19]

First let's get the symbols right:
m1 = larger mass, 4 kg
m2 = smaller mass, 2 kg
T1 = rope tension on the m1 side
T2 = rope tension on the m2 side
r = radius of the pulley, 0.085 m
mp = mass of pulley, 2.6 kg
M = moment or torque due to friction at pulley spindle, 0.51 Nm
J = polar moment of inertia of pulley, 0.5 mp r^2 = ... kg-m^2
a = constant acceleration of the system, till m1 hits ground

Now then, let's start;

at the m1 side, gravity is causing tension T1 after catering to acceleration a, so:
T1 = m1 g - m1 a .........

at the pulley, moment of the tension difference at the center caters to both the frictional torque as well as the angular acceleration of the pulley, which is a/r, hence:
(T1 - T2) r = J a/r + M, or
T1 - T2 = J a/r^2 + M /r = 0.5 mp a + M /r ... 2)

at the m2 side, the torque T2 is simply pulling up mass m2 against gravity at acceleration a, hence:
T2 = m2 g + m2 a ......... 3)

Now, for solving, the simplest way is to eliminate the tensions and get a simple equation in a only, as we are not interested in the other unknowns. The best way is to subtract 3) from 1) and equate that to 2) to eliminate T1 and T2

Once you find a, it's a piece of cake to find the time required to go down a height of 1 m, starting from rest with the equation:
h = 0.5 a t^2, or t = sqrt (2 h/a)

1.58 m/s, 1.78 s

 

[bluebear19]

the above is what i tried to follow, and by following that this is what i arrived at
T1 - T2 = .5mpa +M/r = (m1-m2)g - (m1-m2)a
plugging the numbers in i get:
.5(2.6)a + .51/.085 = (2)(9.8) - 2a
solving for a : a= 4.12 m/s^2
solving for t:
t = sqrt[2ah] = sqrt[2(1) /4.12] = 0.696s
is this right? because the answer given was 1.78s thank you

 

[Hootenanny]

T1 - T2 = .5mpa +M/r = (m1-m2)g - (m1-m2)a

You may want to re-check the highlighted term.

 

[bluebear19]

T1 = m1 g - m1 a
T2 = m2 g + m2 a
then T1 - T2 = (m1-m2)g - (m1-m2)a doesn't it? i don't see what is wrong, can you please tell me?

 

[Hootenanny]

T1 = m1 g - m1 a
T2 = m2 g + m2 a
then T1 - T2 = (m1-m2)g - (m1-m2)a doesn't it? i don't see what is wrong, can you please tell me?

Expanding the brackets:
T1-T2 = m1g - m2g - m1a + m2a

 

[bluebear19]

wouldn't that give the same answer
T1 - T2 = (m1-m2)g - (m1-m2)a = (4-2)(g) - (4- 2)a = 2g - 2a
is the same thing as 4g - 2g - 4a +2a = 2g - 2a

 

[Hootenanny]

wouldn't that give the same answer
T1 - T2 = (m1-m2)g - (m1-m2)a = (4-2)(g) - (4- 2)a = 2g - 2a
is the same thing as 4g - 2g - 4a +2a = 2g - 2a

No. In the post above I expanded your incorrect expression. The correct expression should be

T₁-T₂ = m₁g - m₂g - m₁a - m₂a
=(m₁ - m₂)g - (m₁ + m₂)a

Do you follow?

 

[bluebear19]

oh yes, thank you so much! so the answer should be T1 - T1 = 2g - 4a = .5mpa +M/r
2g - 6a = .5(2.4)a + .51/.085
2g - 6a = 1.2a + 6
a = 1.89
t = 1.03s? does this look right?

 

[Hootenanny]

oh yes, thank you so much! so the answer should be T1 - T1 = 2g - 4a = .5mpa +M/r
2g - 6a = .5(2.4)a + .51/.085
2g - 6a = 1.2a + 6
a = 1.89
t = 1.03s? does this look right?

You're close, just be careful with the mass of the pulley.

 

[bluebear19]

oh i see the mass of the pulley is 2.6 so it should be .5(2.6) this is confusing because in my problem the actualy mass of the pulley is 2.4kg, so other than that everything else is right? thank you sooo much!

 

[Hootenanny]

oh i see the mass of the pulley is 2.6 so it should be .5(2.6) this is confusing because in my problem the actualy mass of the pulley is 2.4kg, so other than that everything else is right? thank you sooo much!

I haven't actually run through the numerical calculations, but the method is certainly correct, as are the initial equations.