Connected particles
Objective. Form simultaneous equations for linked bodies.
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- 2. Worked example
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Learn: the key idea
Connected particles, such as masses joined by a string over a pulley, require writing a separate equation of motion for each particle using F=ma, then solving simultaneously since both particles share the same magnitude of acceleration and tension. For example, two masses m₁ and m₂ connected over a pulley give equations m₁g−T=m₁a and T−m₂g=m₂a, which combine to eliminate T and find a. Always define a consistent positive direction for each particle, usually the direction each one actually moves.
Worked example
Two connected particles of mass 5kg and 3kg hang over a smooth pulley. Form the equation of motion for the 5kg particle, taking downward as positive for it.
- 1Apply F=ma to the 5kg particle
- 2Forces are weight 5g downward and tension T upward
- 3Resultant force equals 5g−T
- 4Equation: 5g−T=5a
Answer: 5g−T=5a
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Start practisingCommon mistake. Students use different values of tension for each particle in a single string system, when tension is the same throughout an inextensible string over a smooth pulley. Always keep T identical in both particles' equations.
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