1. The Atwood Machine
Two masses suspended over a pulley. m2 m1

2. George Atwood
( )
British scientist that designed the Atwood Machine

3. Atwood Machine Free Body Diagram
Newton’s 1st Law
Constant Velocity
ΣF=0
T-W1=0 T-W2=0
T=W1 T=W2
W1=W2
W1=m1g W2=m2g
g=9.8 m/s2 m1 m2 W2 W1

4. Atwood Machine Free Body Diagram
Newton’s 2nd Law
Acceleration
ΣF=ma
T-W1=m1(-a) T-W2=m2(a)
T=W1+m1(-a)
W1+m1(-a)-W2=m2(a)
W1-W2=(m1+m2)(a) or
T=W2+m2a
W2+m2(a)-W1=m1(-a)
W2-W1=(m1+m2)(-a)
W1=m1g W2=m2g
g=9.8 m/s2 m1 m2 a -a W2 W1

5. Alternate Atwood Machine Interpretation

6. Alternate Atwood Machine Interpretation
Newton’s 1st Law
ΣF=0
-W1+W2=0
W1=W2
Newton’s 2nd Law
ΣF=ma
-W1+W2= (m1+m2)a W2>W1
-W1+W2=(m1+m2)(-a) W2<W1 T W1 m1 T W2 m2
constant
velocity
T-W2=0
T-W1=0
W1=W2
acceleration
T-W1=m1(a)
T-W2=m2(a)

7. Pulley and Inclined Planem1 m2

8. Pulley and Inclined Plane Free Body DiagramT f T m1 m2
W|| N W┴ W2
T-W2=0 const vel
T-W2=m2(a) acc up
T-W2=m2(-a) acc down
Newton’s First Law:
ΣF=0 constant velocity (motion or pending motion down)
-W||+f+T=0 -W||+f+W2 =0 (const vel)
Newton’s Second Law:
-W||+f+T=m1(-a) m1 accelerating down the incline
-W||-f+T=m1a m1 accelerating up the incline
The equations can be combined to give: -W||+f+W2=(m1+m2)(-a) (m1 acc down, m2 acc up) W||-f+W2=(m1+m2)a (m1 acc up, m2 down)

9. Alternate Pulley and Inclined Plane Interpretation
W|| m1 f m2 W2
-W||+f+W2 =0 (const vel) -W||+f+W2=(m1+m2)(-a) (m1 acc down, m2 acc up)

10. Pulley Scenario 3 m2 m3 m1

11. T2 T1 T1 m2 m1 T2 m3 W1 W3
Newton’s 1st Law (constant velocity):
T1-W1=0
T1=W1
T2-W3=0
T2=W3
-T1+T2=0
Combining Equations: W1=W3

12. T2 T1 -a T1 m2 m1 T2 -a m3 +a W1 W3
Newton’s 2nd Law (acceleration):
T2-W3=m3(a)
T2=W3+m3(a)
T1-W1=m1(-a)
T1=W1-m1(a)
-T1+T2=m2(-a)
Combining Equations: -W1+m1(a)+W3+m3(a)=m2(-a)
-W1+W3=(m1+m2+m3)(-a)

13. -a m1 m2 m3 W1 W3
Newton’s 1st Law (constant velocity):
Combining Equations: W1=W3
Newton’s 2nd Law (acceleration):
T1-W1=m1(-a)
T1=W1-m1(a)
T2-W3=m3(a)
T2=W3+m3(a)
-T1+T2=m2(-a)
Combining Equations: -W1+m1(a)+W3+m3(a)=m2(-a)
-W1+W3=(m1+m2+m3)(-a)