Suppose that two masses m_{1} and m_{2} 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 - m_{2} g = m_{2} a_{2}

T - m_{1} g =
m_{1} a_{1}
* Constraint: * a

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

Eliminate a_{2}:
2T - m_{2} g = - m_{2} a_{1}/2

T - m_{1} g = m_{1} a_{1}

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

(2 m_{1} - m_{2} ) g =
- (m_{2}/2 + 2 m_{1}) a_{1}

Solution: a_{1} = 2 (
m_{2} - 2 m_{1} )g/(4 m_{1} + m_{2})

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 -
y_{2}) + (h - y_{1})

Differentiate: - 2
v_{2} - v_{1} = 0
→ v_{2} = - v_{1}/2

Steps 3: Energy conservation.

Δ T = - Δ U

m_{1} v_{1}^{2}/2 + m_{2} v_{2}^{2}/2
= - m_{1} g Δ y_{1} - m_{2} g Δ y_{2}

Apply constraint:
m_{1} v_{1}^{2}/2 +
m_{2} v_{1}^{2}/8 = - m_{1} g Δ
y_{1} + m_{2} g Δ y_{1}/2

Solve for v_{1}: v_{1}^{2} = 4 g Δ y_{1}
(m_{2} - 2 m_{1})/ (4m_{1} + m_{2})

From the kinematic equation v^{2} = 2 a Δ x, the above
expression for v_{1} is equivalent to:

a_{1} = 2 ( m_{2} - 2 m_{1} )g/(4 m_{1} +
m_{2})