Chapter 3, Problem 43P

(a)

To determine

The co-efficient of static friction between the floor and the refrigerator.

Answer to Problem 43P

The co-efficient of static friction between the floor and the refrigerator is $0.18$.

Explanation of Solution

Write the expression for maximum force of static friction.

  $F_s={\mu }_{s}N$        (I)

Here, $F_s$ is the static frictional force, ${\mu }_s$ is the co-efficient of static friction between the floor and the refrigerator, and $N$ is the normal force exerted on the refrigerator.

Write the expression for normal force acting on the downward force of gravity on the refrigerator.

  $N=mg$        (II)

Here, $m$ is the mass of the refrigerator, $g$ is the acceleration due to gravity, and $N$ is the normal force.

Substitute equation (II) in equation (I).

  $\begin{array}{c}{F}_s={\mu }_{s}mg\\ {\mu }_s=\frac{F_s}{mg}\end{array}$

Conclusion:

Substitute $350\text{N}$ for $F_s$, $200\text{kg}$ for $m$, and $9.8{\text{m/s}}^2$ for $g$ in above equation to find ${\mu }_s$.

  $\begin{array}{c}{\mu }_s=\frac{350\text{N}}{\left(200\text{kg}\right)\left(9.8{\text{m/s}}^2\right)}\\ =0.18\end{array}$

Therefore, the co-efficient of static friction between the floor and the refrigerator is $0.18$.

(b)

To determine

The co-efficient of kinetic friction between the refrigerator and the floor.

Answer to Problem 43P

The co-efficient of kinetic friction between the refrigerator and the floor is $0.14$.

Explanation of Solution

After the refrigerator starts moving, there is acceleration and the frictional becomes kinetic friction.

Write the expression from Newton’s second law of motion.

  $\begin{array}{c}{\displaystyle \sum F}=ma\\ F_{\text{move}}+F_k=ma\end{array}$        (III)

Here, $F_{\text{move}}$ is the force exerted on the moving refrigerator, $F_k$ is the kinetic frictional force, and $a$ is the acceleration of the refrigerator.

Write the expression for kinetic frictional force.

  $F_k=-{\mu }_{k}mg$        (IV)

Here, ${\mu }_k$ is the co-efficient of kinetic friction between the refrigerator and the floor.

Substitute equation (IV) in equation (III).

  $\begin{array}{c}{F}_{\text{move}}-{\mu }_{k}mg=ma\\ {\mu }_{k}mg=F_{\text{move}}-ma\\ {\mu }_k=\frac{F_{\text{move}}-ma}{mg}\end{array}$        (V)

Write the expression from kinematics equation of motion.

  $v=v_0+at$        (VI)

Here, $v$ is the speed of the refrigerator, $v_0$ is the initial speed, and $t$ is the time period.

Substitute $0$ for $v_0$ and rewrite the above equation.

  $\begin{array}{c}v=at\\ a=\frac{v}{t}\end{array}$        (VII)

Conclusion:

Substitute $2.0\text{m/s}$ for $v$ and $5.0\text{s}$ for $t$ in equation (VII) to find $a$.

  $\begin{array}{c}a=\frac{2.0\text{m/s}}{5.0\text{s}}\\ =0.4{\text{m/s}}^2\end{array}$

Substitute $0.4{\text{m/s}}^2$ for $a$, $350\text{N}$ for $F_{\text{move}}$, $200\text{kg}$ for $m$, and $9.8{\text{m/s}}^2$ for $g$ in equation (V) to find ${\mu }_k$.

  $\begin{array}{c}{\mu }_k=\frac{350\text{N}-\left(200\text{kg}\right)\left(0.4{\text{m/s}}^2\right)}{\left(200\text{kg}\right)\left(9.8{\text{m/s}}^2\right)}\\ =0.14\end{array}$

Therefore, the co-efficient of kinetic friction between the refrigerator and the floor is $0.14$.