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)