1. Definition

A pulley system uses ropes and rotating wheels to transmit forces and change the direction of motion.

In mechanics problems, pulleys are used to analyze:

• tension forces
• connected motion
• acceleration of multiple objects

2. Ideal Pulley Assumptions

In introductory mechanics, pulleys are usually assumed to be:

• massless
• frictionless
• connected by massless ropes that do not stretch

Under these assumptions:

• the tension is the same throughout a single rope
• connected objects share related motion

3. Tension in a Rope

If the rope is ideal:

$$T=\text{constant}$$

throughout the rope.

The tension force always pulls:

• along the rope
• away from the object

4. Single Fixed Pulley

A fixed pulley changes the direction of force but not its magnitude.

Example:

Pulling downward on one side lifts an object upward on the other side.

For an object of mass $m$:

$$\sum F=ma$$

If moving vertically:

$$T-mg=ma$$

5. Two-Object Pulley System (Atwood Machine)

One of the most important pulley systems is the Atwood machine.

Two masses are connected by a rope over a pulley.

Assume:

m2>m1m_2 > m_1m2​>m1​

Then:

• m2m_2m2​ accelerates downward
• m1m_1m1​ accelerates upward

Both masses have the same acceleration magnitude because the rope length remains constant.

6. Newton’s Second Law for Each Mass

For m1m_1m1​:

T−m1g=m1aT – m_1 g = m_1 aT−m1​g=m1​a

For m2m_2m2​:

m2g−T=m2am_2 g – T = m_2 am2​g−T=m2​a

Adding equations eliminates tension:

m2g−m1g=(m1+m2)am_2 g – m_1 g = (m_1 + m_2)am2​g−m1​g=(m1​+m2​)a

Thus:

a=(m2−m1)gm1+m2a = \frac{(m_2 – m_1)g}{m_1 + m_2}a=m1​+m2​(m2​−m1​)g​

7. Tension in the Atwood Machine

Substitute acceleration into either equation:

T=m1(g+a)T = m_1(g+a)T=m1​(g+a)

or

T=m2(g−a)T = m_2(g-a)T=m2​(g−a)

Both give the same result.

8. Connected Motion Constraint

In pulley systems, rope length remains constant.

This creates motion constraints:

• equal displacement magnitudes
• equal velocity magnitudes
• equal acceleration magnitudes

for connected sections of rope.

9. Pulley with Inclined Plane

Pulley systems are often combined with inclines.

Example:

• one mass hangs vertically
• another mass rests on a slope

Then:

For the incline object:

∑F∥=ma\sum F_{\parallel} = ma∑F∥​=ma

where gravity component along slope is:

$$mg\sin \theta$$

Normal force:

$$N=mg\cos \theta$$

Friction may also appear.

10. Free-Body Diagrams

Pulley problems require careful free-body diagrams.

For each object:

• isolate object individually
• draw all forces
• apply Newton’s Second Law separately

Common forces:

• tension
• gravity
• normal force
• friction

11. Physical Importance

Pulley systems illustrate:

• force transmission
• connected acceleration
• constraint relationships
• applications of Newton’s laws

They are foundational for studying:

• tension systems
• rotational dynamics
• mechanical advantage

Summary

In ideal pulley systems:

• tension is constant throughout a rope
• connected objects share related motion

Newton’s Second Law is applied to each object separately:

∑F⃗=ma⃗\sum \vec{F} = m\vec{a}∑F=ma

For an Atwood machine:

a=(m2−m1)gm1+m2a = \frac{(m_2 – m_1)g}{m_1 + m_2}a=m1​+m2​(m2​−m1​)g​

Key ideas:

• analyze each mass independently
• use rope constraints
• apply consistent sign conventions