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202205311107 Problem 4.23

Boxes A and B are in contact on a horizontal, frictionless (i.e. f=0) surface, as shown in the Figure below. Box A has mass 20.0\,\mathrm{kg} and box B has mass 5.0\,\mathrm{kg}. A horizontal force of 100\,\mathrm{N} is exerted on box A. What is the magnitude of the force that box A exerts on box B?

extracted from Problem 4.23, Sears and Zemansky’s University Physics

Steps.

Draw the free-body diagrams of A, B, and A+B.

Apply Newton’s 2^\textrm{nd} law \textrm{Net }F=ma:

\begin{aligned} F_A- {}_{B}F_A - f_A & = m_Aa_A \\ {}_{A}F_B - f_B & = m_Ba_B \\ F_A-f_{A+B} & = m_{A+B}a_{A+B} \\ \end{aligned}

Conditioning the equations of motion:

\begin{aligned} a_A=a_B & =a_{A+B} \\ f_A=f_B=f_{A+B} & =0 \\ {}_{B}F_A & ={}_{A}F_{B} \\ m_A+m_B & =m_{A+B} \\ \end{aligned}

and substituting numbers for symbols, write:

\begin{aligned} 100 - {}_{A}F_{B} & = 20a \\ {}_{A}F_{B} & = 5a \\ 100 & =25a \\ \end{aligned}

\therefore The magnitude {}_{A}F_{B} of the force that box A exerts on box B is 20\,\mathrm{N}.