Finding the force of friction of a moving object and its change when it accelerates to a constant speed

Question

I have been searching for a straight forward answer to this question for ages now and it is driving me crazy.

Here is what I know: If an object is moving at a constant speed the force of friction must equal the applied force (Please assume the applied force is horizontal) and for it to be accelerating or decelerating, the force of friction and the applied force must be uneven.

I know that FK = uK * FN

I have the applied force, the normal force and the coefficient of friction.

This is what I do not understand: If in order to accelerate the applied force must be greater than the friction, how would that work? because surely the object would continue accelerating to infinity? Or does the friction force change to level out and equal the applied force to change the objects state from accelerating to constant velocity? If so, how do i work out the rate of the change in the friction force, surely it does not instantly change otherwise acceleration cannot occur.

Here is a sample situation that may help me understand it better if it is answered

Say I have a box that has a mass of 10 kg and I apply a force horizontally to the box of 50N and the coefficient of kinetic friction is 0.5. How long does it take for the box to finish accelerating and reach a constant velocity? And what is the rate of friction force change?

Then say I suddenly increase the force I am applying to 60N what is the rate of change of the friction force and how long does it take to reach constant velocity?

To recap my questions are: 1. Does the object accelerate to infinity or does the friction force change to equal the applied force? 2. If the friction force does change, how do I work out the rate of change and how long it takes? 3. In the situation above, both times a new force is applied how do I work out change in friction force and how long it takes? Could you also show working so I can understand it properly. 4. If the above is way off and the friction does not change how do I work out the change of velocity over time using friction depending on a change of the applied force?

Sorry for the detailed question I just want to make sure I understand the answer properly :)

PS A simple solution is all I need as I am using this in a simple physics engine for a game.

Thanks, Arch

Answer 1

This might be more detailed than you want; I apologize in advance.

There are two forms of friction:

1. static friction The force of friction exerted on an object when it is at rest.

2. kinetic friction The force of friction exerted on an object when it is in motion.

These two forms of friction have qualitatively properties. Specifically, the force of kinetic friction depends only on the magnitude of the normal force $F_N$ exerted on the moving object and the coefficient of kinetic friction $\mu_k$ of the surface on which it is moving. In fact, as you point at the magnitude of the force of kinetic friction as given by $$ F_k = \mu_k F_N $$ The force of static friction, on the other hand, changes depending on the other external forces on the object.

To understand why, think of a box sitting still on a horizontal table. The box will not feel a friction force in the absence of any other force (if it did, then it would accelerate). However, if you start exerting a small enough force on the box, it still will not move, and in this case, the static friction force is exactly counterbalancing the force you exert. If you push hard enough, however, the box will eventually start sliding. This illustrates that the static friction force can have any value between zero, and some maximum which turns out to be given by $\mu_s F_N$ where $\mu_s$ is the coefficient of static friction. Mathematically, this can be expressed by the following equation: $$ F_s \leq \mu_s F_N $$ where $F_s$ is the magnitude of the static friction force.

Having said all of this, let me reiterate that kinetic friction always has the magnitude $\mu_k F_N$ regardless of the state of motion of the object. If you continuously push an object with a force greater than this value, then it will keep accelerating forever. In order for the acceleration to halt, you would need to reduce the applied force so that it equals the force of static friction.

Lastly, given an applied force $F$, the acceleration of the object will satisfy Newton's second law which says that the net applied force equals the mass of the object multiplied by is acceleration; $$ F - F_k = ma $$ The acceleration of the object is the rate of change of its velocity, so determine the velocity as a function of time you would, in general, have to solve the following differential equation: $$ \frac{dv}{dt} = \frac{1}{m}(F - F_k) $$