91Ó°ÊÓ

The given data can be listed below as,

The mass of crate A is, m_A.

The mass of crate B is, m_B.

The horizontal force is, $\stackrel{→}{\mathrm{F}}$.

The coefficient of kinetic friction is, ${\mathrm{μ}}_{\mathrm{k}}$.

Tension is a pulling force which transmitted axially with the help of rope, string or cable at the end is known as tension.

The expression for frictional force on crate A and b can be expressed as,

$$\begin{gathered} {\mathrm{f}}_{\mathrm{A}}={\mathrm{μ}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{A}}\mathrm{g} \\ {\mathrm{f}}_{\mathrm{B}}={\mathrm{μ}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{B}}\mathrm{g} \end{gathered}$$

Here ${\mathrm{μ}}_{\mathrm{k}}$ is the coefficient of kinetic friction, m_A is the mass of the crate A, m_B is the mass of the crate B.

The expression for force using Newton’s second law can be expressed as,

$\stackrel{→}{\mathrm{F}}-{\mathrm{f}}_{\mathrm{B}}-{\mathrm{f}}_{\mathrm{A}}=0$

Here f_A is the force on crate A, f_B is the force on crate B.

Substitute ${\mathrm{μ}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{A}}\mathrm{g}$ for f_A, ${\mathrm{μ}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{B}}\mathrm{g}$for f_B in above equation.

$$\begin{gathered} \stackrel{→}{\mathrm{F}}-{\mathrm{μ}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{A}}\mathrm{g}-{\mathrm{μ}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{B}}\mathrm{g}=0 \\ \stackrel{→}{\mathrm{F}}={\mathrm{μ}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{m}}_{\mathrm{A}}+{\mathrm{m}}_{\mathrm{B}}\right) \end{gathered}$$

Hence, the required magnitude of $\stackrel{→}{\mathrm{F}}$is, ${\mathrm{μ}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{m}}_{\mathrm{A}}+{\mathrm{m}}_{\mathrm{B}}\right)$.

For Tension considering the block rare one block A and block B as an isolated system, the expression for tension can be expressed as,

$$\begin{gathered} \mathrm{T}-{\mathrm{μ}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{A}}\mathrm{g}=0 \\ \mathrm{T}={\mathrm{μ}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{A}}\mathrm{g} \end{gathered}$$

Hence, the required tension is, ${\mathrm{μ}}_{\mathrm{k}}{\mathrm{m}}_{\mathrm{A}}\mathrm{g}$.