The problem with this question is that static friction and kinetic friction are not fundamental forces in any way-- they're purely phenomenological names used to explain observed behavior. "Static friction" is a term we use to describe the observed fact that it usually takes more force to set an object into motion than it takes to keep it moving once you've got it started.

So, with that in mind, ask yourself how you could measure the relative sizes of static and kinetic friction. If the coefficient of static friction is greater than the coefficient of kinetic friction, this is an easy thing to do: once you overcome the static friction, the frictional force drops. So, you pull on an object with a force sensor, and measure the maximum force required before it gets moving, then once it's in motion, the frictional force decreases, and you measure how much force you need to apply to maintain a constant velocity.

What would it mean to have kinetic friction be greater than static friction? Well, it would mean that the force required to *keep* an object in motion would be greater than the force required to *start* it in motion. Which would require the force to go *up* at the instant the object started moving. But that doesn't make any sense, experimentally-- what you would see in that case is just that the force would increase up to the level required to keep the object in motion, as if the coefficients of static and kinetic friction were exactly equal.

So, common sense tells us that the coefficient of static friction can never be less than the coefficient of kinetic friction. Having greater kinetic than static friction just doesn't make any sense in terms of the phenomena being described.

(As an aside, the static/kinetic coefficient model is actually pretty lousy. It works as a way to set up problems forcing students to deal with the vector nature of forces, and allows some simple qualitative explanations of observed phenomena, but if you have ever tried to devise a lab doing quantitative measurements of friction, it's a mess.)

**Hint** : Friction opposes tendency to move. Tension is produced if the string is stretched $very$ slightly. So, increase friction to maximum and then tension will act if necessary.

Ironically, you are thinking absolutely right. Give yourself a cookie.

From part $a$, we know that the blocks will be at rest at all angles below that.

You are also right as at very small angles there is no need of tension and we can ignore it to solve for, again, an angle condition. You have done excellent work. Congrats.

Now we come to the middle angles. Oh... they drive you insane, don't they?

Let's start. We can start our analysis from 2 blocks, 1 will give a contradiction and other will give a result, but I will start with the one giving contradiction. This will help you.

*All angles are in degrees* :

$\theta=35 $

Lets start by analysing Block A (No racism intended)

Gravity is trying to pull it down : $5*10*\sin(35)N=28.67N$

Friction comes to the rescue(up) : $50*\cos(35)N=16.38N$ // read my hint to know why friction is put max here

As it is at rest, $T=12.29N$

Now Block B is also at rest,

Weight = $114.71N$

max f= $81.92N$

$16.38+114.71=12.29+f$

$f=118.8N$

OOPS, it exceed max value. So, Lets start by analysing Block B. (I love alliteration)

Gravity trying : $114.71N$

Friction comes to the rescue(up) : $81.92N$

You can take from here I guess. calculate tension. Note that you have to revise your calculation for tension again as reaction friction force will be provided by A. Better assume it $f$ from starting FBD of B.

This will yield the correct answer. Friction will be less than max value for upper block. In most cases, You should start analysing with heavier block(my experience). Hope your doubts are cleared.

## Best Answer

Its true that the truck's acceleration does depend on the engine specifications.

However, the maximum acceleration the truck can have is the same as the maximum friction of the wheels from the force of static friction. We don't want the truck wheels to slip. If it accelerates above the static friction force then the wheels will begin to slip. Without this force of friction the truck would not move at all.

$$a=g[\mu_s\cos \theta − \sin \theta]$$ is correct; plugging the values in will give you the maximum acceleration of the truck.

As an additional afterthought to the problem. The fact that the maximum acceleration does not depend on mass implies that it can be said that the maximum acceleration of the truck does not depend on the number of wheels that it has. Assume a truck with 8 wheels (an APC weighing

m1=4 tons), versus a truck with 4 wheels (a SUV weighingm2=1 ton), versus a bike with 2 wheels (m3=20 lbs); the number of wheels reduces the mass parameter used in Newton's second law by a fraction of the total. The m parameter in your equations could be written $$m = m1/8$$, or $$m = m2/4$$, or $$m = m3/2$$ This mass parameter would still cancel out in the end, the APC, the SUV, and the Bike all would have the same maximum acceleration.