Atwood’s Machine

Example

Blocks A and B of mass 6 kg and 2 kg respectively are connected by a light inextensible rope hung over a light smooth pulley. Calculate the tension in the rope.

atwood1 Solution

Since the rope is inextensible, block A will have the same downward acceleration as block B’s upward acceleration. Also the tension forces must be between 2g and 6in order for block A and B to experience a net upward and downward force respectively

atwood2

  • Considering both blocks as a combined 8 kg mass,

atwood3

\displaystyle \begin{aligned}({{F}_{net}}&=ma)\\6g-2g&=(6+2)a\text{ }\\a&=0.5g\end{aligned}

  • Considering block A by itself as a 6 kg mass,

atwood4

\displaystyle \begin{aligned}({{F}_{net}}&=ma)\\6g-T&=(6)(0.5g)\text{ }\\T&=3.0g\end{aligned}

  • Alternatively, we can consider blocks B by itself as a 2 kg mass.

atwood5

\displaystyle \begin{aligned}({{F}_{net}}&=ma)\\T-2g&=(2)(0.5g)\\T&=3.0g\text{ }\end{aligned}

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