01 · Light
What is neglected when a string is modelled as light?
Hint
Compare with the masses it connects.
Worked solution
Its mass. That does not mean its tension or length is zero.
Understand · explore · practise
Strings, pulleys, beads and pegs
Separate light, taut, inextensible and smooth assumptions, and use a fixed-pulley string-length constraint without overgeneralising.
Before you startThe mechanics modelling cycle and object models; signed changes.
01 / Name each assumption separately
A useful string model needs more than one adjective.
Light concerns mass; inextensible concerns length.
Taut means stretched tight along the modelled path, rather than slack. Smooth contact means friction is neglected. These statements answer different questions.
One fixed pulley, one taut stringExplore
The two straight vertical parts initially each have length 1 m. The fixed curved part stays unchanged. Increase the left straight length by d and keep the total string length fixed.
02 / A string can pull but does not push
Tension acts along the string away from the attached body.
02 · Inextensible
What is held fixed by an inextensible-string assumption?
Hint
Consider stretching.
Worked solution
The string length does not change. This does not by itself say the string is taut or that every connected object moves along the same direction.
03 · Slack
Can a slack ideal flexible string push its two attached objects apart?
Hint
A string does not transmit compressive thrust.
Worked solution
No. An ideal string can exert tension when taut, but cannot push. If it is slack, a taut-string motion constraint must not be applied.
03 / Write the constraint before inferring motion
The pulley is fixed and the contact arc is unchanged.
Let a and b be the two vertical straight lengths and c the fixed curved length.Worked example
a+b+c=L
The total inextensible string length L is fixed.
Δa+Δb=0
Because c is fixed, increasing one straight part decreases the other equally.
Δa=+0.3 m ⇒ Δb=−0.3 m
With downward positive on each side, one end goes down while the other goes up.
Watch: fixed string length links the two displacements
Pause, replay or seek freely. The notes explain the same idea and stay in view.
04 · Other side
If the left end moves down by 0.45 m, what happens to the right end in this arrangement?
Hint
Keep the sum of the two straight lengths unchanged.
Worked solution
The right end moves up by 0.45 m while the string remains taut and the geometry stays as specified.
05 · Sign convention
If both sides use downward-positive coordinates, are their displacement changes equal or opposite?
Hint
Translate the direction words into signs.
Worked solution
They are opposite: Δb=−Δa. Their magnitudes are equal.
04 / Equal magnitudes are not equal vectors
Keep the geometry attached to the statement.
06 · Velocity
For differentiable taut-string motion in this fixed arrangement, relate the signed vertical velocities using downward positive on both sides.
Hint
Differentiate a+b=constant with respect to time.
Worked solution
da/dt+db/dt=0, so the signed velocities are opposite and their speed magnitudes are equal.
07 · Acceleration
State the corresponding acceleration relation under the same conditions.
Hint
Differentiate the constraint again.
Worked solution
The signed vertical accelerations sum to zero. They have equal magnitudes but opposite vertical directions when nonzero.
08 · Changed geometry
Why should you not automatically reuse this relation for a moving pulley with extra supporting string segments?
Hint
Count the lengths that change.
Worked solution
The length constraint can contain more than two changing segments, with different coefficients. Derive that arrangement’s own constraint before comparing motions.
05 / Equal tension uses force assumptions
Inextensibility alone is not the justification.
09 · Ideal pulley
In the standard model of one continuous light string over a smooth light pulley, what tension simplification is normally used?
Hint
Ignore string mass and pulley friction/inertia.
Worked solution
The tension magnitude is the same along that ideal string. Its direction changes with the string path; the two forces on the connected bodies need not point in the same direction.
10 · Rough pulley
If friction at the pulley matters, may the two tensions differ?
Hint
Contact can transmit tangential force.
Worked solution
Yes. The equal-tension idealisation is no longer generally justified. Use the physical assumptions and an appropriate force model.
11 · Compression error
A proposed string calculation gives negative tension. What should be checked?
Hint
An ideal string cannot push.
Worked solution
Check the sign convention, equations and whether the assumed taut configuration is possible. A physically negative tensile magnitude indicates the assumed model or configuration is unsuitable.
06 / Beads, wires and pegs constrain motion
Smooth means no frictional contact force.
12 · Smooth straight wire
A bead slides on a smooth straight wire. In what direction is the contact reaction relative to the wire?
Hint
The smooth contact supplies no tangential friction.
Worked solution
It is perpendicular to the wire; the bead is constrained to the wire’s path. This does not mean the reaction is always vertical.
13 · Peg
What does a smooth peg idealisation omit where a string passes over it?
Hint
Distinguish friction from the contact force itself.
Worked solution
Friction at the contact is neglected. A normal contact interaction can still change the string’s direction.
07 / Check when the idealised arrangement changes
A condition can cease to hold during motion.
14 · Endpoint
Why stop using the simple two-hanging-mass diagram once one mass reaches the floor?
Hint
A new contact can alter the forces and the tautness.
Worked solution
The floor introduces another interaction and the string may become slack. The original motion and force assumptions must be reconsidered.
The model shown here is one fixed pulley with two vertical straight string parts and an unchanged contact arc. It is not a universal diagram for every system involving a string.
08 / Translate words into precise consequences
Geometry gives constraints; force assumptions give force relations.
Light: neglect mass. Inextensible: fixed total length. Taut: the stated path constraint is active. Smooth: no friction at contact. Derive motion relations from the actual path, and use equal tension only when the ideal force assumptions support it.
Section 1 of 8 · Name each assumption separately