Chapter 4, Problem 64A

To determine

The normal forces between the top, middle, and bottom blocks.

Answer to Problem 64A

The normal force between the top block and the middle block is $45\text{N}$ .

The normal force between the bottom block and the middle block is $57\text{N}$ .

The normal force between the bottom block and the surface is $93\text{N}$ .

Explanation of Solution

Given info:

Consider three blocks are stacked on top of one another.

Consider the top, middle, and bottom blocks are denoted by $T,M,B$ .

The mass of the top block is $m_{\text{T}}=4.6\text{kg}$ .

The mass of the middle block is $m_{\text{M}}=1.2\text{kg}$ .

The mass of the bottom block is $m_{\text{B}}=3.7\text{kg}$ .

Consider the acceleration due gravity of the Earth is $g=9.80{\text{m/s}}^2$ .

Formula used:

Newton’s Second Law:

The object’s acceleration is equivalent to the sum of the forces acting on the object divided by the object’s mass. That is,

  $\begin{array}{l}a=\frac{F_{\text{net}}}{m}\\ F_{\text{net}}=ma\end{array}$

Calculation:

Consider the normal forces exerted by the top block on the middle block is denoted by $F_{\text{MT}}$ .

Consider the normal forces exerted by the middle block on the top block is denoted by $F_{\text{TM}}$

Consider the normal forces exerted by the bottom block on the middle block is denoted by $F_{\text{BM}}$ .

Consider the normal forces exerted by the middle block on the bottom block is denoted by $F_{\text{MB}}$ .

Consider the normal forces exerted by the surface on the bottom block is denoted by $F_{\text{SB}}$ .

Consider the normal forces exerted by the Earth’s mass on top, middle, and bottom block is denoted by $F_{\text{ET}},F_{\text{EM}},F_{\text{EB}}$ respectively.

Consider the normal force is between the top and the middle blocks.

Consider the top block is the system and upward direction as positive.

  $\begin{array}{l}{F}_{\text{net}}=F_{\text{MT}}-F_{\text{ET}}\\ ma=F_{\text{MT}}-F_{\text{ET}}\\ m\left(0\right)=F_{\text{MT}}-F_{\text{ET}}\\ F_{\text{MT}}=F_{\text{ET}}\end{array}$

  $\begin{array}{l}{F}_{\text{MT}}=m_{\text{T}}g\\ F_{\text{MT}}=4.6\times 9.80\text{kg}\cdot {\text{m/s}}^2\times \left(\frac{\text{1}\text{N}}{\text{1}\text{kg}\cdot {\text{m/s}}^2}\right)\\ F_{\text{MT}}=45\text{N}\end{array}$

Thus, the normal force between the top block and the middle block is $45\text{N}$ .

Consider the normal force is between the bottom and the middle blocks.

Consider the middle block is the system and upward direction as positive.

  $\begin{array}{l}{F}_{\text{net}}=F_{\text{BM}}-F_{\text{TM}}-F_{EM}\\ ma=F_{\text{BM}}-F_{\text{TM}}-m_{\text{M}}g\\ m\left(0\right)=F_{\text{BM}}-F_{\text{TM}}-1.2\times 9.80\text{kg}\cdot {\text{m/s}}^2\times \left(\frac{1\text{N}}{\text{1}\text{kg}\cdot {\text{m/s}}^2}\right)\\ 0=F_{\text{BM}}-F_{\text{TM}}-11.76\end{array}$

As the force $\left(F_{\text{TM}}\right)$ exerted by the top block on the middle block is equal to the force $\left(F_{\text{MT}}\right)$ exerted by the middle block on the top block.

  $\begin{array}{l}0=F_{\text{BM}}-F_{\text{MT}}-11.76\\ 0=F_{\text{BM}}-45-11.76\\ F_{\text{BM}}=56.76\text{N}\\ F_{\text{BM}}\approx 57\text{N}\end{array}$

Thus, the normal force between the bottom block and the middle block is $57\text{N}$ .

Consider the normal force is between the bottom block and the surface.

Consider the bottom block is the system and upward direction as positive.

  $\begin{array}{l}{F}_{\text{net}}=F_{\text{SB}}-F_{\text{MB}}-F_{E\text{B}}\\ ma=F_{\text{SB}}-F_{\text{MB}}-m_{\text{B}}g\\ m\left(0\right)=F_{\text{SB}}-F_{\text{MB}}-3.7\times 9.80\text{kg}\cdot {\text{m/s}}^2\times \left(\frac{1\text{N}}{\text{1}\text{kg}\cdot {\text{m/s}}^2}\right)\\ 0=F_{\text{SB}}-F_{\text{MB}}-36.26\end{array}$

As the force $\left(F_{\text{BM}}\right)$ exerted by the bottom block on the middle block is equal to the force $\left(F_{\text{MB}}\right)$ exerted by the middle block on the bottom block.

  $\begin{array}{l}0=F_{\text{SB}}-F_{\text{BM}}-36.26\\ 0=F_{\text{SB}}-57-36.26\\ F_{\text{SB}}=93.26N\\ F_{\text{SB}}\approx 93N\end{array}$

Thus, the normal force between the bottom block and the surface is $93\text{N}$ .

Conclusion:

The normal force between the top block and the middle block is $45\text{N}$ .

The normal force between the bottom block and the middle block is $57\text{N}$ .

The normal force between the bottom block and the surface is $93\text{N}$ .