ENG1011 Chap.3 Springs and Pulley Systems
Springs and Pulley Systems
Springs and Pulley Systems applies Week 1 equilibrium to two common devices in Week 2. A linear spring obeys Hooke's law, carrying a force equal to its stiffness times its change in length, so a spring is both a force carrier and a way of measuring force through stretch. Springs that share the same two end bars sit in parallel: they stretch equally, their forces add, and their stiffnesses add.
Springs joined end to end sit in series: each carries the full force, their stretches add, and the reciprocals of their stiffnesses add. Any network can be reduced to one equivalent spring by working from the innermost group outward. Ideal pulleys are frictionless and weightless, so one continuous rope carries one tension along its length and each pulley only redirects it.
Cutting every rope segment that supports a moving block shows that the block's load is shared equally among those segments, which is the source of a pulley system's mechanical gain.
Combined problems put a spring somewhere in a pulley system, so the rope force comes from counting segments and the stretch from the spring.
Week 2 questions usually give a network diagram and ask for one number: an equivalent stiffness as a multiple of k, a single spring's extension, a rope tension, or the mass a system can hold. The practice test includes two spring items and one pulley item of this kind.
Because the answers are auto-marked, the arithmetic has to be exact, so keep fractions until the last line and check that the units of stiffness and load agree before dividing.
What this chapter covers
- 01
Hooke's law for a linear spring
- 02
Parallel springs: shared stretch, added stiffness
- 03
Series springs: shared force, added flexibility
- 04
Reducing a spring network to one equivalent stiffness
- 05
Ideal pulleys and uniform rope tension
- 06
Counting the segments that support a moving block
- 07
Fixed pulleys, brackets and the loads they carry
- 08
Combining a spring and a movable pulley
Worked example · free
Equivalent stiffness of three identical springs
- 1The side-by-side pair is in parallel: 300 + 300 = 600 N/m.
- 1That pair is in series with the third spring: 1/ke = 1/600 + 1/300 = 3/600, so ke = 200 N/m.
- 1Total stretch: 60/200 = 0.30 m, made of 0.10 m in the pair and 0.20 m in the single spring.
Key terms
- Spring stiffness
- The force needed per unit change in length of a spring, measured in newtons per metre.
- Springs in parallel
- Springs that share both end connections, so they stretch equally and their stiffnesses add.
- Springs in series
- Springs joined end to end, so each carries the full force and their stretches add.
- Ideal pulley
- A frictionless, weightless pulley that changes the direction of a rope without changing its tension.
- Supporting rope segment
- A length of rope that pulls on a moving block, sharing the block's load with the other segments.
- Equivalent stiffness
- The stiffness of a single spring that would replace a whole spring network under the same load.
Springs and Pulley Systems FAQ
How can I tell whether springs are in series or parallel?
Look at what they share. Springs fixed to the same rigid bar at both ends must stretch by the same amount, so they are in parallel. Springs connected end to end, with nothing else attached between them, carry the same force, so they are in series.
Why does a pulley reduce the force needed to lift a load?
Several rope segments pull up on the moving block at once, and each carries the same tension. The load divides among them, so the force needed at the free end equals the load divided by the number of supporting segments, at the cost of pulling more rope.
Does a fixed pulley add to the mechanical gain?
No. A pulley attached to the ceiling only turns the rope through an angle. Count only the segments attached to the block that moves with the load. The fixed pulley's bracket still has to carry the rope tensions on both sides of it.
Should spring answers be left in terms of k?
When the springs are given as multiples of k, keep k as a symbol until the end. Results like 2k or 6k/13 are exact and easy to check, while decimals hide the structure of the network.
Exam move
Build fluency by redrawing spring networks one reduction at a time, replacing each parallel group by its sum before applying the series rule. For pulleys, trace the rope from its anchored end to its free end and tick every segment that meets the moving block. Combined problems reward doing the force step and the geometry step separately and in that order.
After each spring question, check that the forces in a parallel group add back to the load and that the stretches in a series chain add to the total.
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