10.9 Friction between horizontal surfaces (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 friction between horizontal surfaces. The questions are categorized in terms of the characterizing features of the subject matter :

• An external force on the block
• An external force on the plank
• One external force each on block and plank
• Block with initial velocity

An external force on the block

Problem 1 : In the arrangement shown in the figure, the mass of the pulleys and strings are negligible. The block “B” rests on the smooth horizontal surface, whereas the coefficient of friction between two blocks is 0.5. If masses of the blocks “A” and “B” are 2 kg and 4 kg respectively, then find the maximum force for which there is no relative motion between two blocks.

Solution : According to the question, two blocks have no relative motion between them. The friction force is less than limiting friction. As the limiting case, the maximum friction between the blocks is equal to limiting friction :

$$F_S=\mu N=\mu m_{A}g=0.5X2X10=10N$$

Since blocks have no relative motion between them, they move as a single unit with a common acceleration, say “a”. The vertical and horizontal forces by the two brackets holding pulleys balance each other. Hence, these forces are not considered. The free body diagram of the combined body is shown on the left side of the figure. The friction forces between the blocks are internal forces for the combined body and as such not considered. Note that we have not drawn components of forces in vertical direction as the analysis in vertical direction is not relevant here.

$$\Rightarrow a=\frac{F}{\left(m_A+m_B\right)}=\frac{F}{6}$$

$$\Rightarrow F=6a$$

Considering free body diagram of block “B” as shown on the right side of the figure, we have :

$$a=\frac{F_S}{m}=\frac{10}{4}=2.5m/s^2$$

Putting in the expression of force, we have,

$$\Rightarrow F=6X2.5=15N$$

An external force on the plank

Problem 2 : In the arrangement shown in the figure, mass of block and plank are 2 kg and 3 kg respectively. The coefficient of friction between block and plank is 0.5, whereas there is no friction between plank and underneath horizontal surface. An external force of 50 N is applied on the plank as shown. If the length of plank is 5 m, find the time after which block and plank are separated.

Solution : First, we need to know the nature of the friction between block and plank. If the friction is less than liming friction, then two entities will move together with an acceleration given by :