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Friction between horizontal surfaces

Module by: Sunil Kumar Singh

Summary: Friction plays the role of both cause and moderator of motion.

Friction plays different roles in motion. Consider the case of a block on a plane slab (plank), which, in turn, is placed on a horizontal surface. Typically, friction moderates the motion of a body. In other words, friction retards motion. On the other hand, friction also acts as the “cause” of the motion under certain circumstance. In the role of a “moderator”, friction negates external force. In the role of the “cause” of motion, it is either the sole external force or the greater external force on the body responsible for its motion.

In the "block - plank" set up, we can apply force either on the block or on the plank or on both of them simultaneously. The resulting motion depends on varieties of factors such as friction, external force and location of application of external force etc.

Figure 1: The block and plank move with respect to each other.
Friction between horizontal surfaces
Friction between horizontal surfaces (fb1.gif)

In order to analyze situation as mentioned above, we need to have clear understanding of the way friction works on each of bodies. In this module, we seek to organize ourselves so that we have well thought out plan and method to deal with this kind of motion.

First thing that we need to know is about friction – its magnitude and direction. Secondly, we need to know the nature of friction – whether friction is static, limiting or kinematic. Once, we have complete picture of various friction forces at different interfaces, we are in position to draw the free body diagram and analyze the motion.

Friction between interfaces

We shall consider three cases. In order to keep the matter simple, we make one simplifying assumption that the friction between plank and the horizontal surface is negligible. In other words, the underlying horizontal surface is smooth.

  • An external force is applied on the block.
  • An external force is applied on the plank.
  • One external force each is applied on block and plank.

To keep the description uniform, we refer block as “A”, having mass, “m”, and plank as “B”, having mass, “M”. We shall find that a consistency in denoting block and plank is very helpful in analyzing motion. The analysis of motion for first two cases is similar. It differs only to the extent that point of application of external force changes. Nevertheless, it is interesting to analyze two cases separately to score the differences in two cases.

An external force is applied on the block

The external force is applied on the block as shown in the figure below. There are two possibilities : (i) there is no friction between "A" and "B" or (ii) there is friction between "A" and "B".

Figure 2: An external force, "F", is applied on the block.
An external force on the block
An external force on the block (fb2.gif)

There is no friction between block and plank

We first analyze the motion for a case when all interfaces are “friction – less”. In this situation, the external force on block, “A”, accelerates only the block. As there is no friction between the interface of block and plank, there is no external force on the plank in horizontal direction. As such, plank, “B”, is not accelerated. The free body diagrams of “A” and “B” are as shown in the figure.

Figure 3: There is no external force on the plank.
Free body diagrams
Free body diagrams (fb3.gif)

The acceleration of block, “ a A a A ”, is :

a A = F m a A = F m

The acceleration of plank, “ a B a B ”, is :

a B = 0 a B = 0

There is friction between block and plank

The friction force in this case will influence the motion as it acts as external force on each individual body. The resulting motion of the bodies, however, would depend on the nature of friction (static, limiting or kinetic). In the figure below, the forces on the block and plank are shown separately for each of them.

Figure 4: Friction is the only external force on the plank.
Forces on the bodies
Forces on the bodies (fb4.gif)

In order to determine the direction of frictions at the interface, we go by two simple steps. We first consider the body on which external force (F) is applied. The direction of friction on the body (A) is opposite to the external force. We determine friction on the other body (B) by applying Newton’s third law of motion. Friction on plank (B) is equal in magnitude, but opposite in direction.

Once direction of friction is known, we need to know the nature of friction. For this, we first calculate limiting friction between the bodies so as to compare it with external applied force "F". The limiting friction is given by :

F S = μ N = μ m g F S = μ N = μ m g

Important thing to realize here is that limiting friction depends only on the mass of the block, “m”, and is independent of the mass, “M”, of the plank. Now, we should compare external force with friction to determine its magnitude. Finally, we analyze motion in following manner :

1 : F ≤ F S F ≤ μ s m g F ≤ F S F ≤ μ s m g

In this case, friction self adjusts to external force, “F”. Hence, static friction is given by :

⇒ f s = F ⇒ f s = F

As there is no relative motion, the block and plank move together as a single unit. The friction forces at the interface are internal forces for the combined body. The free body diagram of the combined body is shown here.

Figure 5: Free body diagram of combined body.
Free body diagram
Free body diagram (fb5.gif)

The common acceleration of the block and plank, “a”, is :

a = a A = a B = F m + M a = a A = a B = F m + M

2 : F > F S F > μ s m g F > F S F > μ s m g

The block and plank move with different accelerations and have relative motion. The friction between the interfaces is, therefore, kinetic friction.

The free body diagrams of block and plank are shown here.

Figure 6: Free body diagrams of block and plank.
Free body diagrams
Free body diagrams (fb6.gif)

The acceleration of block, “ a A a A ”, is :

a A = F - μ k m g m a A = F - μ k m g m

The acceleration of block, “ a B a B ”, is :

a B = μ k m g M a B = μ k m g M

An external force is applied on the plank

The external force is applied on the plank as shown in the figure below. There are two possibilities : (i) there is no friction between "A" and "B" or (ii) there is friction between "A" and "B".

Figure 7: An external force, "F", is applied on the plank.
An external force on the plank
An external force on the plank (fb7.gif)

There is no friction between block and plank

In this situation, the external force on block, “B”, accelerates only the plank. As there is no friction between the interface of block and plank, there is no external force on the block in horizontal direction. As such, block, “A”, is not accelerated. The free body diagrams of “A” and “B” are as shown in the figure.

Figure 8: There is no external force on the block.
Free body diagrams
Free body diagrams (fb8.gif)

The acceleration of block, “ a A a A ”, is :

a A = 0 a A = 0

The acceleration of plank, “ a B a B ”, is :

a B = F M a B = F M

There is friction between block and plank

The friction force in this case will influence the motion as it acts as external force on each individual body. The resulting motion of the bodies, however, would depend on the nature of friction (static, limiting or kinetic). In the figure below, the forces on the block and plank are shown separately for each of them.

Figure 9: Friction is the only external force on the block.
Forces on the bodies
Forces on the bodies (fb9.gif)

In order to determine the direction of frictions at the interface, we go by two simple steps. We first consider the body on which external force is applied. The direction of friction on the body (B) is opposite to the external force. We determine friction on the other body (A) by applying Newton’s third law of motion. Friction on block (A) is equal in magnitude, but opposite in direction.

Once direction of friction is known, we need to know the nature of friction. Unlike in the previous case when external force is applied on the block, the situation here is different. The plank carries another mass of block over itself. The external force (F) is not completely used to overcome friction at the interface, but to move the combined mass together. As such, external force can not be directly linked to static friction as in the case when force is applied on the block.

For this reason, we shall adopt a different strategy. We shall assume that friction is static friction. If the analysis of force does not support this assumption, then we correct the assumption accordingly. Now, the limiting friction is given by :

F S = μ N = μ m g F S = μ N = μ m g

Let us assume that " f S f S " be the static friction between the surfaces. As there is no relative motion, the block and plank move together as a single unit. The friction forces at the interface are internal forces for the combined body. The free body diagram of the combined body is shown here.

Figure 10: Free body diagram of combined body.
Free body diagram
Free body diagram (fb5.gif)

The common acceleration of the block and plank, “a”, is :

a = a A = a B = F m + M a = a A = a B = F m + M

We observe that friction is the only force on the block. Then, applying Newton's second law for the motion of block, we have :

f S = m a f S = m a

Clearly, if " f S f S " so calculated is less than or equal to limiting friction, then block and plank indeed move together. However, if " f S f S " is greater than limiting friction, then block and plank move with different accelerations. In such case, friction between "A" and "B" is kinetic friction.

The free body diagrams of block and plank are shown here.

Figure 11: Free body diagrams of block and plank.
Free body diagrams
Free body diagrams (fb10.gif)

The acceleration of block, “ a A a A ”, is :

a A = μ k m g m = μ k g a A = μ k m g m = μ k g

The acceleration of block, “ a B a B ”, is :

a B = F - μ k m g m a B = F - μ k m g m

One external force each is applied on block and plank

In this case, external forces act on the block and plank separately. It is, therefore, not possible in this case to compare limiting force with external force as there are two of them, which are acting on two different bodies.

Figure 12: The block and plank move with respect to each other.
Friction between horizontal surfaces
Friction between horizontal surfaces (fb1.gif)

Here, we adopt the strategy of assuming nature of friction before hand. First, we calculate the limiting friction as before,

F S = μ N = μ m g F S = μ N = μ m g

Then, we consider a friction, “ f S f S ”, which is static friction, but whose magnitude is not known. If we are wrong in our assumption as brought out by the analysis, then we will correct our assumption; otherwise not. The free body diagrams of the block and the plank are shown in the figure.

Figure 13: Free body diagrams of block and plank.
Free body diagrams
Free body diagrams (fb11.gif)

Since friction is static unknown friction, two bodies move together without any relative motion between them. It means that :

a A = a B a A = a B

F B − f s M = f s − F A m F B − f s M = f s − F A m

⇒ m F B − m f s = M f s − M F A ⇒ m F B − m f s = M f s − M F A

⇒ f s m + M = m F B + M F A ⇒ f s m + M = m F B + M F A

⇒ f s = m F B + M F A m + M ⇒ f s = m F B + M F A m + M

If the value of “ f s f s ” as calculated, using expression derived above, is less than limiting friction, then we conclude that two bodies are indeed moving together. Otherwise, our assumption were wrong and the bodies are moving with different accelerations. In that case, we calculate accelerations separately as :

⇒ a A = f s − F A m ⇒ a A = f s − F A m

⇒ a B = F B − f s M ⇒ a B = F B − f s M

Velocity .vs. Force

We have so far analyzed motion, considering external force on the bodies. Under certain condition, if data is provided in the form of velocity of the individual body, then the analysis is simplified significantly. Consider the set up as shown in the figure. At a certain instant, the block is imparted a velocity (alternatively, we have simply allowed a block with certain velocity to slide over the plank underneath).

Figure 14: The block and plank move with respect to each other.
Friction between horizontal surfaces
Friction between horizontal surfaces (fb12.gif)

An initial velocity to block, here, ensures that there is relative motion at the interface. This, in turn, ensures that the friction between the surfaces is kinetic friction. This means that nature of friction is known and need not be investigated as in the case when external force is applied.

Example

Problem : A block with initial velocity "v" is placed over a rough horizontal plank of the same mass. The plank resides over a smooth horizontal plane. Plot the velocities of the block and the plank with respect to time.

Figure 15: Initial velocity of block is "v".
Block and plank system
Block and plank system (fb14.gif)

Solution : Let the mass of the block and plank each be “m” and “μ” be the coefficient of kinetic friction between them. Since block is given a velocity with respect to ground, the friction between block and plank is kinetic friction, given by :

F K = μ N = μ m g F K = μ N = μ m g

The direction of friction on the block is opposite to the direction of motion. The friction here retards the motion. Let us denote block and plank by subscripts “1” and “2” respectively.The deceleration of the block is given by :

a = μ m g m = μ g a = μ m g m = μ g

Figure 16: Friction on two bodies are equal but opposite.
Forces on the bodies
Forces on the bodies (fb15.gif)

It means that velocity of the block decreases at constant rate with time. The velocity – time plot of block, therefore, is a straight line with negative slope. On the other hand, friction is the only force on the plank in the forward direction. The magnitude of acceleration is same as that of block, because it has the same mass and it is worked by force of same magnitude. Since plank starts from zero velocity, the velocity – time plot of the plank is a straight line of constant slope, starting from the origin of the plot.

The block looses motion due to deceleration, whereas plank acquires motion due to acceleration. A situation comes when speeds of the two entities are equal. In that situation,

v 1 = v 2 v 1 = v 2

v − a t = 0 + a t v − a t = 0 + a t

t = v 2 a = v 2 μ g t = v 2 a = v 2 μ g

The common velocity is given by :

v c = v 2 = a t = μ g X v 2 μ g = v 2 v c = v 2 = a t = μ g X v 2 μ g = v 2

This means that velocity of the block and plank becomes equal to half of the initial velocity imparted to the block. There is no relative motion between two entities. They simply move together with same velocity as combined mass with the common velocity, “v/2”. The required plot is shown in the figure :

Figure 17: Bodies acquire common velocity after some time.
Velocity - time plot
Velocity - time plot (fb16.gif)

It must also be understood that the above approximates an ideal condition; the plank and block will eventually stop as there is some friction between plank and the underneath horizontal surface.

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