10.3 Working with friction (application)

Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

Representative problems and their solutions

We discuss problems, which highlight certain aspects of the study leading to working with friction. The questions are categorized in terms of the characterizing features of the subject matter/ question :

• Equilibrium
• Acceleration along incline
• Motion with friction
• Contact force on an incline

Equilibrium

Stack of blocks

Problem 1 : The coefficient of friction between sand particles of uniform size is “μ”. Find the maximum height of the sand pile that can be made in a circular area of radius “r”.

Solution : We consider a sand particle on the slant side of the sand pile. We observe here that the particle on the sand pile resembles the situation that of the motion of a particle on an incline of certain angle. The pile will have the maximum height for which the friction between the particles has maximum friction i.e. limiting friction. Beyond this angle, the sand particle will slide down and spill outside the circular base boundary of the sand pile.

We know that the angle of incline corresponding to the limiting friction is the angle of repose,

$$\tan \theta =\mu$$

From the geometry of right angle ABC, we have :

$$\tan \theta =\mu =\frac{AB}{BC}=\frac{h_{\max }}{r}$$

$$\Rightarrow h_{\max }=\mu r$$

Problem 2 : A homogeneous chain of length “L” lies on a horizontal surface of a table. The coefficient of friction between table and chain is “μ”. Find the maximum length of the chain, which can hang over the table in equilibrium.

Solution : Here, we have to consider the translational equilibrium of the chain for which net force on the chain should be zero. Now, the chain is homogeneous. It means that the chain has uniform linear mass density. Let it be “λ”. Further, let a length “y” hangs down the table in equilibrium. The length of chain residing on the table, then, is "L – y".

The mass of the chain,” $m_1$ ”, on the table is

$$\Rightarrow m_1=\left(L-y\right)\lambda$$

The mass of the chain,” $m_2$ ”, hanging is

$$m_2=y\lambda$$

Free body diagram of the mass of chain, “ $m_1$ ”, on the table is shown in the figure above. This mass is pulled down by the weight of the hanging chain. For balanced force condition, the friction force on the chain on the table should be equal to the weight of the hanging chain,

$$\mu m_{1}g=m_{2}g$$

$$\Rightarrow \mu \left(L-y\right)\lambda g=y\lambda g$$

$$\Rightarrow \mu \left(L-y\right)=y$$

$$\mu L-\mu y=y$$

$$y\left(1+\mu \right)=\mu L$$

$$\Rightarrow y=\frac{\mu L}{\left(1+\mu \right)}$$

Problem 3 : In the arrangement of three blocks on a smooth horizontal surface, the friction between blocks “A” and “B” is negligible, whereas coefficient of friction between blocks “B” and “C” is “μ”. Find the minimum force “F” to prevent middle block from sliding down.