Difference between revisions of "Forest UCM NLM AtwoodMachine"

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:<math>a_r =</math> acceleration of <math>m_1</math> with respect to the lower pulley
 
:<math>a_r =</math> acceleration of <math>m_1</math> with respect to the lower pulley
  
with respect to the earth
+
assuming that <math>a_1</math> is moving upwards with respect to the earth
  
 
:<math>a_1 = a_r - a_3</math>  : <math>a_3 =</math> acceleration of lower pully as well as <math>m_3</math>
 
:<math>a_1 = a_r - a_3</math>  : <math>a_3 =</math> acceleration of lower pully as well as <math>m_3</math>
Line 71: Line 71:
 
similarly
 
similarly
  
:<math>a_2=-a_r-a_3</math>
+
:<math>a_2=-a_r-a_3</math> : if <math>m_1</math> is accelerating upwards then <math>m_2</math> is accelerating downwards
  
 
==Step 5: Use Newton's second law==
 
==Step 5: Use Newton's second law==

Revision as of 11:54, 22 August 2014

Simple Atwood's machine

TF UCM SAM 1.gif


[math]\Rightarrow T = \frac{2m_1m_2}{m_1+m_2} g[/math]

Double Atwood's machine

TF UCM DAM 1.gif


The problem

Determine the acceleration of each mass in the above picture.

Step 1: Identify the system

Each block is a separate system with two external forces; a gravitational force and the rope tension.

Step 2: Choose a suitable coordinate system

A coordinate system with one axis that defines the posive direction as up is one possible orientation.

Step 3: Draw the Free Body Diagram

200 px

Step 4: Define the Force vectors using the above coordinate system

for mass 1
[math]T_1 - m_1 g = m_1 a_1[/math]
for mass 2
[math]T_2 - m_2 g = m_2 a_2[/math]
for mass 3
[math]T_3 - m_3 g = m_3 a_3[/math]


If we know the mass of all the objects in the system then we are left with three unkown Tensions and three unknown acceleratios. In total we currently have 6 unkowns and 3 equations.


Using Newton's third law we know that [math]T_1 = T_2[/math] reducing the unkowns to 5.

We need 2 more equations!

External Forces on Lower pulley

Consider the external forces acting on the MASSLESS lower pulley


[math]T_3-T1-T2 =T_3-T1-(T1) =(0)a[/math]
[math]T_3=2T_1[/math]


Now we have 4 unkwons and 3 equations

relative acceleration

let

[math]a_r =[/math] acceleration of [math]m_1[/math] with respect to the lower pulley

assuming that [math]a_1[/math] is moving upwards with respect to the earth

[math]a_1 = a_r - a_3[/math] : [math]a_3 =[/math] acceleration of lower pully as well as [math]m_3[/math]


similarly

[math]a_2=-a_r-a_3[/math] : if [math]m_1[/math] is accelerating upwards then [math]m_2[/math] is accelerating downwards

Step 5: Use Newton's second law

Forest_UCM_NLM#Atwoods_Machine