# How can an object with no applied force by an external agent still have friction?

To begin, this is what I've learned about how to calculate the friction for any case:

• kinetic friction acting on a body is the Frictional coefficient x Normal acting on the body
• however, if kinetic friction > applied external force (i.e. without friction, ofc), then the object is static w.r.t. the observer, and static friction = applied external force

Now here comes my doubt:
We give a body, on a rough horizontal surface, a force for 5 seconds. Now this external force was successful in overcoming the static friction, and thus the body has some net external force and acceleration. However, after 5 seconds the external force applied is ceased, and so is the friction (since it is a self-adjusting force). Therefore, after 5 seconds, the net acceleration is zero, but at 5 seconds, it still does retain some velocity. Now, Newton's first law of motion states that a body in uniform motion will retain its state and magnitude of motion until and unless and unless external force is applied. Now, as per our observations in the practical world, the object must stop. However, there is no friction since there is no applied force. But friction does oppose relative motion between any two objects in contact. So, either what I know about friction is less than required, or some concept of mine has to fixed.

• There doesn’t have to be an “applied” force for there to be friction. There just needs to be a force. In the example you gave the normal force is still present, and is what is responsible for friction. Commented Apr 24 at 13:46

That is not true. As long as there is nonzero relative velocity between the two surfaces, the friction will be constant and equal to the kinetic friction. The friction only exactly opposes the applied force when the relative velocity is zero and the applied force is less than the static friction threshold. Both conditions need to be true.

• I got the static friction part, but you say that as long as there is relative motion, the friction = frictional coefficient x Normal?? Commented Apr 23 at 11:34
• @AyushSingh Yes Commented Apr 23 at 11:58
• so as the velocity tends to zero, the friction force remains constant, until the velocity reaches zero. A non-continuous graph in mechanics? really? Even if its true, I'm not convinced. can you give more examples or anything related to it? Commented Apr 23 at 12:01
• @AyushSingh Yes, the friction becomes zero when the velocity becomes zero provided there is no external force. Commented Apr 23 at 12:05
• @AyushSingh Yes, it is indeed discontinuous in the ideal case. It is zero at $v=0$ and $+\mu_k N$ for $v\lt 0$ and $-\mu_k N$ for $v\gt 0$. It's a bump function. Commented Apr 23 at 12:24

however, if kinetic friction > applied external force (i.e. without friction, ofc), then the object is static w.r.t. the observer, and static friction = applied external force

Kinetic friction can never be greater than the applied force. It can less than the applied force, in which case there will be a net force on the object and the object will accelerate. Once the object is in motion the applied force can be reduced to exactly equal the friction force for a net force of zero and the object will continue moving at constant speed. In general, kinetic friction does not occur until the applied force exceeds the maximum possible static friction force.

However, after 5 seconds the external force applied is ceased, and so is the friction (since it is a self-adjusting force). Therefore, after 5 seconds, the net acceleration is zero, but at 5 seconds, it still does retain some velocity.

The kinetic friction force does not immediately cease when the applied force is removed. It is not a self adjusting force. It is generally assumed to remain constant as long as the object is in motion. The self adjusting force is the static friction force which matches the applied force up until the maximum possible static friction force is reached and kinetic friction takes over.

When the applied force is removed, the kinetic friction force now becomes the only external force acting on the object causing it to decelerate until it comes to a stop.

Therefore, after 5 seconds, the net acceleration is zero, but at 5 seconds, it still does retain some velocity.

After the applied force is removed the acceleration becomes negative due to the kinetic friction force that opposes its motion, not zero. It will have an initial velocity equal to that it at at the instant the external force is removed, then decelerate to zero.

Now, Newton's first law of motion states that a body in uniform motion will retain its state and magnitude of motion until and unless and unless external force is applied.

That's what the law says, but when the applied force is removed the external force becomes the kinetic friction force alone causing the body to decelerate.

EXAMPLE;

The following example will sum up the above. Refer to the friction plot shown in the figure below taken from the following website, which admits to it being simplistic : http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html#kin

Let's assume I have a large box of mass $$m$$ resting on a horizontal surface (floor) with friction. Let the coefficients of static and kinetic friction be $$\mu_s$$ and $$\mu_k$$, respectively, where $$\mu_{s}\gt \mu_k$$ as is generally the case.

I begin to apply a pushing force, but the box does not move because there is now a static friction force equal and opposite to my pushing force. As I gradually increase my pushing force the static friction increases to oppose my force, continuing to prevent the box from moving. We say this tells us that the static friction force is "self-adjusting". In terms of the friction plot, we are on the line where the static friction force exactly equals the applied force.

I continue to increase my applied force until I eventually reach the point where my applied force equals the maximum possible static friction force of $$\mu_{s}mg$$. If you have ever tried to push something on a floor, at this point you might experience it suddenly "breaking free" with a drop in the resistance to your pushing. In terms of the friction plot this is idealized as a step function (in reality there may be some transition).

Now the friction becomes kinetic equal to $$\mu_{k}mg$$, which is lower than the maximum static friction force, and which acts opposite to the motion of the box. The friction plot shows the force to be constant implying it is independent of the applied force as well as the velocity of the object. In reality this is only approximately true and only for limited ranges of velocities. For our "textbook" example, we will assume it is.

The box is now in motion. If I continue to apply the force that caused it to move (the maximum static friction force), and the kinetic friction force remains constant, there will now be a net force $$F_{net}$$ of

$$F_{net}=\mu_{s}mg-\mu_{k}mg\tag{1}$$

From Newtons second law the acceleration of the box will be

$$a=\frac{F_{net}}{m}=\frac{mg(\mu_{s}-u_{k})}{m}=g(\mu_{s}-\mu_{k})\tag{2}$$

Assuming the acceleration remains constant, the velocity of the box $$v_{0}$$ at some time $$t_{0}$$ (which could be 5 s as in your example) will be $$at_{0}$$, or

$$v_{0}=g(\mu_{s}-\mu_{k})t_{0}\tag{3}$$

At time $$t_{0}$$ I stop pushing the box. The box will continue to slide due to its inertia but its motion is opposed by the kinetic friction force, which is now the only horizontal force acting on the box, giving it a negative acceleration of

$$a=-\mu_{k}g\tag{4}$$

From kinematics the velocity of the box as a function of time is now

$$v(t)=v_{0}-\mu_{k}gt\tag{5}$$

The time it takes the box to stop, i.e., when $$v(t)=0$$ and kinetic friction ceases is then theoretically

$$t=\frac{v_{0}}{\mu_{k}g}\tag{6}$$

Note that if instead of removing my force at time $$t_{0}$$ I reduced it to exactly equal the kinetic friction force, the net force would be zero and the box would continue to move at constant velocity of $$v_{0}$$ per Newton's first law.

Bottom Line; Static friction only exists in opposition to an applied force to prevent the relative motion between contacting surfaces. It matches the applied force with an upper limit. Kinetic friction only exists in opposition to relative motion already underway between contacting surfaces. That motion may or may not be due to an applied force parallel to the surface.

Hope this helps.

• This is also what I thought, but how to calculate it? Commented Apr 24 at 1:42
• @AyushSingh what is it you want to calculate? Commented Apr 24 at 5:12
• @AyushSingh I will include an example of pushing a box on a rough surface with calculations Commented Apr 24 at 9:57
• @AyushSingh See the example now in my answer. Sorry if it's a bit long. Commented Apr 24 at 13:18
• @AyushSingh I've showed you the calculations. Do you have any further questions? Also, not the bottom line statement at the end of my answer. I believe that answers your questions. Commented Apr 25 at 22:01

How can an object with no applied force by an external agent still have friction?

The coefficient of friction is a ratio of the normal force on a surface to the tangential friction force resisting motion. Typically the normal force is due to gravity so that the frictional force is

$$F_{friction} = \mu mg$$ where $$\mu$$ is the coefficient of friction. So friction exists without an external tangential force but certainly requires an external normal force which is provided by gravity.

however, if kinetic friction > applied external force (i.e. without friction, ofc), then the object is static w.r.t. the observer

This is not correct. If the object is static w.r.t to the friction surface, then there is no kinetic friction, so kinetic friction cannot be > applied external force. kinetic friction requires the object to be moving wrt to the friction surface and this creates heat. If the object is stationary w.r.t the fiction surface then the friction is static friction. Caution is required for describing the friction between a rolling object and the surface. The friction in this case is static friction as the contact patch with the surface is not moving w.r.t to the surface. Rolling friction is usually less than kinetic friction, which is why the bearings of car wheels usually have ball bearings, where the balls roll inside their cages, rather than just having a bush bearing where the circular surfaces slide relative to each other.

Now, as per our observations in the practical world, the object must stop. However, there is no friction since there is no applied force.

When a block is sliding along a surface, there is a kinetic friction force, regardless of whether or not a force is applied. The equation for kinetic friction is $$\mu \ m g$$ where $$\mu$$ is the coefficient of friction, and $$mg$$ is the force normal to the surface. This equation is independent of the applied force parallel to the surface and also independent of the velocity of the sliding object relative to the surface (surprisingly), other than the requirement that the relative velocity be non zero. When the applied horizontal accelerating force is removed, the (constant) kinetic friction continues to act and slows the object down.

However, after 5 seconds the external force applied is ceased, and so is the friction (since it is a self-adjusting force).

This is also not correct. Dynamic friction is not a self adjusting force as far as the velocity or accelerating force acting on the object parallel to the surface is concerned. As long as the force of gravity acting normal to the surface is constant and as long as the velocity is non zero, the dynamic friction does not care about the parallel force or velocity. (If dynamic friction was self adjusting and always equal to the force trying to accelerate the object parallel to the surface, there would be no net force and no acceleration.)