Unit 2.7 | Problem Solving – Simple Pulley Systems

The basics

The most basic pulley system, also referred to as an Atwood’s machine, consists of 2 masses connected by a rope. The tension is the same throughout the entire rope. The rope goes around a massless pulley as shown below.

And what if the pulley had mass? Then we’d need “Torque” (a force’s turning effect) to spin it, which we will cover much later in Unit 5.

The direction of acceleration

The pulley system will always accelerate in the direction of the heavier mass.

Solving Pulley Problems

There are two ways we can solve pulley problems.

1. Using our force framework covered in previous sections (the long way)
2. Solving as a system (the easy way)

Method one: draw an FBD for each mass (like the ones above) and write Newton’s second law for each, then solve the two equations together. Method two: treat both masses as one system. The tension is internal, so the net force is [katex] (m_1 – m_2)g [/katex] acting on the total mass [katex] m_1 + m_2 [/katex], which gives [katex] a = \dfrac{(m_1 – m_2)g}{m_1 + m_2} [/katex].

PQ – Pulleys (Intermediate)

1. Two objects (49.0 and 24.0 kg) are connected by a massless string that passes over a massless, frictionless pulley. The pulley hangs from the ceiling. Find the acceleration of the objects and the tension in the string.
2. A window washer pulls herself upward using the bucket–pulley system. She sits in the bucket, and when she pulls down on the rope the bucket moves up.
   • How hard must she pull downward to raise herself slowly at constant speed?
   • If she increases this force by 15%, what will her acceleration be? The mass of the person plus the bucket is 72 kg.

PQ – Pulleys (Advanced)

1. Block A (m_A) of mass 13kg is held in place on a frictionless table. It is connected by a massless rope to Block B (m_B) of mass 5kg, that is hanging off the table.
   • Find the acceleration of each block when the system is released from rest.
   • Now if Block A is released from rest 1.25m from the edge of the table. How long does it take to get there?
2. Let’s replicate the situation above. This time, however, the table has a coefficient of static friction of .4 (µ_s = .4) and a kinetic friction of .2 (µ_k = .2).
   • What minimum value of m_A will keep the system from starting to move?
   • What value of m_A will keep the system moving at constant speed?

3. A block of mass m₁ (5kg) is on a ramp (µ_k = .22) that is inclined at 20° above the horizontal. It is connected by a massless string to a block of mass m₂ (8kg) that hangs over the top edge of the ramp.
   • What is the acceleration of the masses and the tension in the string?
   • If m₁ = 8 kg, and the coefficient of static friction between m₁ and the incline is 0.300, find the maximum value of m₂ that will not make m₁ slide up the incline.

Lesson 2.7 Recap

In this lesson you learned how to solve pulley problems in two different ways.

Lesson 2.8 Preview

In the next lesson, we will take “solving problems as a system” to the next step. We will cover how to solve problems where there are 2 or more objects — similar to what we did with the pulley system. ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-2-7-advanced-problem-solving-pulley-systems/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-2-7-advanced-problem-solving-pulley-systems/