Static and Kinetic Friction

Question

A couch has a mass of 138 kg, and between the couch and the floor there's a coefficient of static friction of 0.516 and a coefficient of kinetic friction of 0.264 If the lurch lasts 5.83 milliseconds, how fast in meters per second do you and the couch move?

I would appreciate if you can at least help me set up the equation.

Answer 1

The wording of the question suggests a missing context providing more

information about the lurch and how it came about.

So I will have to make some assumptions discussed below.

Let

m = 138 kg be the mass of the couch,

f_s = 0.516 be the coefficient of static friction,

f_k = 0.264 be the coefficient of kinetic friction,

t = 5.83 ms be the duration of the lurch,

v (to be computed) be the speed after at the end of the lurch,

g = 9.81 m/s² be gravitational acceleration.

During the lurch the couch is subject to 2 horizontal forces:

F_p is the force of pushing in the direction of movement,

F_f is the force of kinetic friction opposite the direction of movement.

Therefore the net force ΣF has magnitude

ΣF = F_p - F_f

We can use Newton's second law in the form

ΣI = Δp, where

ΣI is the impulse of the net force, and

Δp is the change in momentum.

The change in momentum Δp = mv, because the initial speed is 0.

ΣI = I_p - I_f, where

I_p is the impulse of the pushing force F_p, and

I_f is the impulse of force of kinetic friction F_f.

The force of kinetic friction

F_f = mgf_k

and it remains constant; therefore

I_f = mgf_kt

So the equation you are looking for is of the form

I_p - mgf_kt = mv

From that you could calculate v if we knew I_p.

To calculate the impulse I_p of F_p, we need some information about F_p,

which I assume is in the missing context.

Most likely, you are trying to push the couch with a larger and larger force until it starts moving.

In that case the couch starts moving with the minimal force sufficient to overcome static friction;

this minimal force is

F_p = mgf_s.

There is the other alternative that right from the beginning you try to push the couch with a much larger force.

This alternative is unlikely for two reasons --

we would have no information about the pushing force F_p, and

secondly the given coefficient of kinetic friction would be irrelevant.

Having decided on the initial value of F_p the next problem is to figure out

how the pushing force changes during the given lurch.

In the absence of any information about it, I can think of two alternatives.

1) One possibility is that the force F_p remains constant during the lurch.

In that case

I_p = F_pt = mgf_st

and the equation you need for calculating v is

mgf_st - mgf_kt = mv. (1)

2) The other alternative is that the lurch duration is not just an arbitrary number,

but instead, in the missing context we are told that the force F_p decreases during the lurch,

so that it becomes 0 at the end of the lurch.

If this is the right assumption then the question is how it decreases.

And in the absence of any information the only reasonable assumption is that it decreases linearly.

That would make

I_p = mgf_st/2

and the equation you need for calculating v is

mgf_st/2 - mgf_kt = mv (2)

Both equations (1) and (2) can be simplified:

gt(f_s - f_k) = v (3)

gt(f_s/2 - f_k) = v (4)

The first thing to notice is that both (3) and (4) are independent of the mass of the couch,

which puts into question my assumptions.

The other thing is that the given value of f_s is a bit smaller than double f_k,

which would still make the speed v negative in equation (4) --

that is not physically possible, which implies that assumption 2) must be rejected.

The bottom line is that only those assumptions leading to equation (3) seem possible.