Effect of friction on a moving body under a variable force

Question

Given the system in the figure:

The block slides on a horizontal surface and is acted upon by a force $P$ which varies in magnitude as shown. Knowing that the coefficients of friction between the block and the surface are $\mu_s = 0.6$ and $\mu_k = 0.25$ and that the block is initially at rest, I have to determine the velocity of the block at $t  =  5  s$.

I'm using the principle of linear impulse and momentum to solve the problem. $$mv_1 + \sum \int fdt= mv_2$$ What I want to know is how to split the frictional force. Should I use $f_r=\mu_sN$ once $P$ drops below $\mu_sN$ or once it drops below $\mu_kN$. Also, what happens once $P$ is between $\mu_sN$ and $\mu_kN$, will the object remain in motion or will it stop?

What I mean is summarized in this equation: $$mv_1 + \int P dt -\mu_kN \times t_1 -\mu_sN \times (5-t_1) = mv_2$$ where $t_1$ is the time at which $P$ drops below $\mu_kN$.

Answer 1

First, you need to review the behavior of frictional forces. The simplest model for static friction is : $$\left|\mathcal{F}_s\right| \le \mu_s F_N$$ if the object is not sliding. If another force pushes with a magnitude greater than $\mu_s F_N$, the object will slide and the static friction no longer exists. It will not ``reappear'' until the object stops sliding.

For kinetic friction $$\left|\mathcal{F}_k\right| = \mu_k F_N$$ when an object is sliding (and the entire time it's sliding). The direction of this force is opposite the direction of sliding. If the normal force magnitude is constant, this magnitude is constant until the object stops. It is independent of any other forces parallel to the surface.

Bottom line: 1) Is the initial force great enough to produce sliding? If yes, then static friction never appears until the object stops. 2) If the pushing force is great than the kinetic friction, the object will gain speed. 3) If the pushing force is less than the kinetic friction, the object will lose speed. 4) The kinetic friction is constant in this problem.

My suggestion is to find if the total impulse (from P and friction) ever returns to zero before the P force stops.

Answer 2

What I want to know is how to split the frictional force. Should I use fr=μsN once P(t) drops below μsN or once it drops below μkN. Also, what happens once P is between μsN and μkN, will the object remain in motion or will it stop?

Once a body starts moving in accelerated motion the kinetic coefficient of friction has to be taken into consideration.

The coefficient of static friction can tell you the limiting pull necessary to start the motion. The static coefficient can be used only when it is preventing the motion.

Hint: At t=0 the driving force is 100 N and at that moment the magnitude of frictional force will be less than the driving force.

Therefore the system starts with initial velocity 0 under the influence of a force {P(t)- frictional force using kinetic frictional coefficient }.

The total impulse can be calculated from P(t) -time graph alongwith negative contribution from friction.

frictional force , which is a constant can be calculated from the use of coefficient of kinetic friction as the body is moving.