Applying Newton's Laws — Question 10

Question 10

A maximum cord tension bounds an allowed acceleration

Two blocks A and B, masses 2 kg and 4 kg, are connected by a light horizontal cord on a frictionless table. A force P pulls A away from B. The cord can sustain at most 12 N. Initially both are at rest.

1. Find acceleration and tension as functions of nonnegative P.

2. Find the largest P that keeps the cord within its limit.

3. If that largest P is applied for 2 s, find speed and displacement while the cord remains intact.

Question 10 - Solution

A maximum cord tension bounds an allowed acceleration

(a) Use the complete system and then B.

$$P=(2+4)a\ \Rightarrow\ a=P/(6\,\mathrm{kg}),\quad T=(4\,\mathrm{kg})a=\tfrac23P$$ The total mass is 6 kg, so $a=P/(6\ \mathrm{kg})$. B is pulled only by the cord, giving $T=4a=(2/3)P$.

(b) Apply the connector bound.

$$\tfrac23P\le12\,\mathrm N\ \Rightarrow\ \boxed{0\le P\le18\,\mathrm N},\quad a_{\max}=3\,\mathrm{m/s^2}$$ $(2/3)P\le12$ N gives $0\le P\le18$ N. At the upper limit, $a=3$ m/s$^2$. A 12 N cord limit does not mean the external pull is limited to 12 N.

(c) Evolve the intact system.

$$v=at=3(2)=6\,\mathrm{m/s},\quad\Delta x=\tfrac12at^2=\tfrac12(3)(2)^2=6\,\mathrm m$$ At constant $a=3$ m/s$^2$ for 2 s, $v=6$ m/s and $\Delta x=6$ m. This prediction assumes no break, stop or other interaction invalidates the model during the interval.