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The Atwood Machine m1m1 m2m2 Two masses suspended over a pulley.

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George Atwood (1746-1807) British scientist that designed the Atwood Machine

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Atwood Machine Free Body Diagram m1m1 m2m2 W1W1 T W2W2 T Newton’s 1 st Law Constant Velocity ΣF=0 T-W 1 =0 T-W 2 =0 T=W 1 T=W 2 W 1 =W 2 W 1 =m 1 g W 2 =m 2 g g=9.8 m/s 2

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Atwood Machine Free Body Diagram m1m1 m2m2 W1W1 T -a W2W2 T Newton’s 2 nd Law Acceleration ΣF=ma T-W 1= m 1 (-a) T-W 2 =m 2 (a) T=W 1 +m 1 (-a) W 1 +m 1 (-a)-W 2 =m 2 (a) W 1 -W 2 =(m 1 +m 2 )(a) or T=W 2 +m 2 a W 2 +m 2( a)-W 1 =m 1 (-a) W 2 -W 1 =(m 1 +m 2 )(-a) W 1 =m 1 g W 2 =m 2 g g=9.8 m/s 2 a

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Alternate Atwood Machine Interpretation m1m1 m2m2 m1m1 m2m2 w1w1 w2w2

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m1m1 m2m2 w1w1 w2w2 Newton’s 1 st Law ΣF=0 -W 1 +W 2 =0 W 1 =W 2 Newton’s 2 nd Law ΣF=ma -W 1 +W 2 = (m 1 +m 2 )a W 2 >W 1 -W 1 +W 2 =(m 1 +m 2 )(-a) W 2

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Pulley and Inclined Plane m1 m2

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Pulley and Inclined Plane Free Body Diagram m1 m2m2 N W┴W┴ W || T Newton’s Second Law: -W || +f+T=m 1 (-a) m 1 accelerating down the incline -W || -f+T=m 1 a m 1 accelerating up the incline f T-W 2 =0 const vel T-W 2 =m 2 (a) acc up T-W 2 =m 2 (-a) acc down The equations can be combined to give: -W || +f+W 2 =(m 1 +m 2 )(-a) (m 1 acc down, m 2 acc up) -W||-f+W 2 =(m 1 +m 2 )a (m 1 acc up, m 2 down) T W2W2 Newton’s First Law: ΣF=0 constant velocity (motion or pending motion down) -W||+f+T=0 -W||+f+W 2 =0 (const vel)

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Alternate Pulley and Inclined Plane Interpretation m1m1 m2 W || W2 f -W || +f+W 2 =0 (const vel) -W || +f+W 2 =(m 1 +m 2 )(-a) (m 1 acc down, m 2 acc up)

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Pulley Scenario 3 m2m2 m1m1 m3m3

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m1m1 m3m3 m2m2 T1-W 1 =0 T1=W 1 -T 1 +T 2 =0 T 2 -W 3 =0 T 2 =W 3 T1T1 W1W1 T1T1 T2T2 T2T2 W3W3 Combining Equations: W 1 =W 3 Newton’s 1 st Law (constant velocity):

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m1m1 m3m3 -a m2m2 +a T 1 -W 1 =m 1 (-a) T 1 =W 1 -m 1 (a) -T 1 +T 2 =m 2 (-a) T 2 -W 3 =m 3 (a) T 2 =W 3 +m 3 (a) T1T1 W1W1 T1T1 T2T2 T2T2 W3W3 Combining Equations: -W 1 +m 1 (a)+W 3 +m 3 (a)=m 2 (-a) -W 1 +W 3 =(m 1 +m 2 +m 3 )(-a) Newton’s 2 nd Law (acceleration):

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m1m1 m3m3 m2m2 -a T 1 -W 1 =m 1 (-a) T 1 =W 1 -m1(a) -T 1 +T 2 =m 2 (-a) W1W1 W3W3 Combining Equations: -W 1 +m 1 (a)+W 3 +m 3 (a)=m 2 (-a) -W 1 +W 3 =(m 1 +m 2 +m 3 )(-a) Newton’s 2 nd Law (acceleration): Newton’s 1 st Law (constant velocity): T 2 -W 3 =m 3 (a) T 2 =W 3 +m 3 (a) Combining Equations: W 1 =W 3

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