Chapter 7, Problem 7.24P

A uniform plank is supported by a fixed support at A and a drum at B that rotates clockwise. The coefficients of static and kinetic friction for the two points of contact are as shown. Determine whether the plank moves from the position shown if (a) the plank is placed in position before the drum is set in motion; and (b) the plank is first placed on the support at A and then lowered onto the drum, which is already rotating.

To determine

(a)

Whether the plank moves from the position if the plank is placed in position before the drum is set in motion

Answer to Problem 7.24P

The co-efficient of friction ${\mu }_A$ at point A is larger than the co-efficient of static friction ${\left({\mu }_S\right)}_A$. Therefore, the plank will slide at point A.

Explanation of Solution

Given information:

A is fixed support.

Drum rotates at clockwise.

If there is no relative motion between two surfaces that are in contact, the relationship between normal force $N$ and friction force $F$ would be:

  $F={\mu }_{s}N$

  ${\mu }_s$ - Co-efficient of static friction

Steps to follow in the equilibrium analysis of a body are:

1. Draw the free body diagram.

2. Write the equilibrium equations.

3. Solve the equations for the unknowns.

Calculation:

FBD of plank

Assume $N_A,N_B$ as the reaction force at point A and B respectively.

Assume $F_A,F_B$ as the friction force at point A and B.

For the equilibrium of plank, the bending moment about point A is equal to zero.

  ${\displaystyle \sum M_A=0}$

  $\begin{array}{l}3N_B-2W=0\\ N_B=\frac{2W}{3}\end{array}$

Write equilibrium equation in horizontal direction.

  $\to {\displaystyle \sum F_x}=0$

  $\begin{array}{l}{F}_B-F_A=0\\ F_A=F_B\end{array}$

Write equilibrium equation in vertical direction.

  $\uparrow {\displaystyle \sum F_y}=0$

  $\begin{array}{l}{N}_A+N_B-W=0\\ N_A=\frac{W}{3}\end{array}$

The plank to remain at rest, it must resist the maximum static friction force $F_B$ at point B.

Assume ${\left({\mu }_s\right)}_B$ as the co-efficient of static friction at point B.

  $\begin{array}{l}{F}_B={\left({\mu }_s\right)}_B{N}_B\\ =0.18\left(\frac{2W}{3}\right)\\ =0.12W\end{array}$

Therefore

  $F_A=F_B=0.12W$

The co-efficient of static friction ${\mu }_A$ at point A

  $\begin{array}{l}{\mu }_A=\frac{F_A}{N_A}\\ =\frac{0.12W}{\left(\frac{W}{3}\right)}\\ =0.36\end{array}$

  ${\mu }_A>{\left({\mu }_S\right)}_A$

Therefore, the plank will slide at point A.

Conclusion:

The co-efficient of friction ${\mu }_A$ at point A is larger than the co-efficient of static friction ${\left({\mu }_S\right)}_A$. Therefore, the plank will slide at point A.

To determine

(b)

Whether the plank moves from the position if the plank is first placed on support A and then lowered onto the drum

Answer to Problem 7.24P

The co-efficient of friction ${\mu }_A$ at point A is smaller than the co-efficient of static friction ${\left({\mu }_S\right)}_A$. Therefore, the plank will not move.

Explanation of Solution

Given information:

A is fixed support.

Drum rotates at clockwise.

If there is no relative motion between two surfaces that are in contact, the relationship between normal force $N$ and friction force $F$ would be:

  $F={\mu }_{s}N$

  ${\mu }_s$ - Co-efficient of static friction

Steps to follow in the equilibrium analysis of a body are:

1. Draw the free body diagram.

2. Write the equilibrium equations.

3. Solve the equations for the unknowns.

Calculation:

According to sub part a

  $N_A=\frac{W}{3}$

  $N_B=\frac{2W}{3}$

The plank to remain at rest, it must slip on the drum at point B.

  $\begin{array}{l}{F}_B={\left({\mu }_k\right)}_B{N}_B\\ =0.15\left(\frac{2W}{3}\right)\\ =0.1W\end{array}$

Therefore

  $F_A=F_B=0.1W$

The co-efficient of static friction ${\mu }_A$ at point A

  $\begin{array}{l}{\mu }_A=\frac{F_A}{N_A}\\ =\frac{0.1W}{\left(\frac{W}{3}\right)}\\ =0.3\end{array}$

  ${\mu }_A<{\left({\mu }_S\right)}_A$

Therefore, the plank will not move.

Conclusion:

The co-efficient of friction ${\mu }_A$ at point A is smaller than the co-efficient of static friction ${\left({\mu }_S\right)}_A$. Therefore, the plank will not move.