Worked Example

Suppose that two masses m1 and m2 are connected to the pulley system shown in the transparency. What is the acceleration of each mass? What is the speed of each mass?

Find acceleration by DID TASC.

Step 1: Draw diagram and coordinate system(s).
Step 2: Isolate the system(s).
Step 3: Draw all forces.
Step 4: Take components.
Step 5: Apply Newton's second law for each system. Apply constraints.
Step 6: Solve.
Step 7: Check.

Step 5:     2T - m2 g = m2 a2
T - m1 g = m1 a1
Constraint:
a2 = - a1/2

This gives 3 equations and 3 unknowns → problem is solvable.

Eliminate a2: 2T - m2 g = - m2 a1/2
T - m1 g = m1 a1

Eliminate T (multiply 2nd equation by 2 and subtract from the first):

(2 m1 - m2 ) g = - (m2/2 + 2 m1) a1

Solution:   a1 = 2 ( m2 - 2 m1 )g/(4 m1 + m2)

Alternatively, find speed by D0EL.

Step 1: Define/draw diagram and coordinate system.
Step 2: Choose a consistent zero.
Step 3: Energy conservation.
Step 4: Losses.

Steps 1 & 2: Define coords and consistent zero.

Equation of constraint:   L = 2 (h - y2) + (h - y1)
Differentiate:   - 2 v2 - v1 = 0   →   v2 = - v1/2

Steps 3: Energy conservation.

Δ T = - Δ U
m1 v12/2 + m2 v22/2 = - m1 g Δ y1 - m2 g Δ y2

Apply constraint:  m1 v12/2 + m2 v12/8 = - m1 g Δ y1 + m2 g Δ y1/2
Solve for v1:   v12 = 4 g Δ y1 (m2 - 2 m1)/ (4m1 + m2)

From the kinematic equation v2 = 2 a Δ x, the above expression for v1 is equivalent to:
a1 = 2 ( m2 - 2 m1 )g/(4 m1 + m2)