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Mechanics · Forces & Newton's laws

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Connected particles on slopes and pulleys

Two particles joined by a light inextensible string over a smooth pulley — one on a slope, one hanging — share the same acceleration and string tension; write F = ma along each particle's line of motion (including the slope component and friction) and solve simultaneously.

Mechanics · Forces & Newton's laws

Connected particles on slopes and pulleys

Two particles joined by a light inextensible string over a smooth pulley — one on a slope, one hanging — share the same acceleration and string tension; write F = ma along each particle's line of motion (including the slope component and friction) and solve simultaneously.

Why it works

This is where everything comes together: an inclined plane, friction, and connected particles in one problem. Two facts make it tractable. Because the string is inextensible, both particles have the same acceleration (in magnitude). Because it is light and the pulley smooth, the tension is the same throughout the string. So one unknown acceleration aa and one unknown tension TT describe the whole system.

The method is always the same: write Newton's second law along each particle's line of motion, taking the direction of motion as positive.
  • Hanging particle (mass MM): Mg−T=MaMg - T = Ma.
  • Particle on the slope (mass mm, angle α\alpha): resolve along the slope. Its
weight contributes mgsin⁡αmg\sin\alpha down the slope; the normal reaction is R=mgcos⁡αR = mg\cos\alpha, so friction (if rough) is up to μR\mu R, opposing motion. If it moves up the slope, T−mgsin⁡α−F=maT - mg\sin\alpha - F = ma.

That's two equations in aa and TT — add them to eliminate TT and find aa, then substitute back for TT. Always decide first which way the system moves (compare the hanging weight MgMg with the slope pull mgsin⁡αmg\sin\alpha, plus friction).