Friction problem would really appreciate the help

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The discussion revolves around a physics problem involving a box on a truck that accelerates, leading to the box potentially falling off. The coefficients of friction indicate that the box will slide since the required force to keep it stationary exceeds the maximum static friction. The kinetic friction force acting on the box is calculated to be 19.0 N, which is the only horizontal force influencing its motion. By analyzing the box's acceleration and the truck's movement, the time it takes for the truck to catch up to the box is determined to be approximately 1.03 seconds. Understanding the transition from static to kinetic friction is crucial for solving similar friction-related problems.
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Homework Statement


Losing Cargo. A box of mass 12.1 kg rests on the flat floor of a truck. The coefficients of friction between the box and floor are mu_s = 0.190 and mu_k = 0.160. The truck stops at a stop sign and then starts to move with an acceleration of 2.18 m/s^{2}.

If the box is a distance 1.83 m from the rear of the truck when the truck starts, how much time elapses before the box falls off the truck?

Homework Equations


f_smax= mu_s * N
f_k = mu_k * N
f_s + f_k = ma

x-x_0 = v_0x + 1/2 (a_x)t^2

The Attempt at a Solution


f_smax = .190(12.1*9.8) = 22.5

f_k = .160(12.1*9.8) = 19.0

22.5 + 19.0 =12.1(a)
41.5/12.1 = a = 3.43

1.83 = 1/2(3.43)t^2

t=1.03 seconds
 
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i'm totally lost

i'm having a really difficult time with friction problems any guidance would be really appreciated
 
Important thing to note... the frictional force is pushing the box forward here...

The first question that needs to be answered is, does the box slide?

Calculate the force required to keep the box from sliding... ie assume the box isn't sliding and get the frictional force...

frictional force = ma = 12.1*2.18 = 26.378N

But the max. static frictional force is 22.5N as you calculated...

26.378N is beyond 22.5N so the box must be sliding...

So now we can ignore static friction and deal with kinetic friction.

You've calculated the frictional force = 19.0N... this is the only force acting on the box horizontally... hence 19.0N = ma

get the acceleration of the box...

now we have a kinematics problem.

the box is ahead of the rear of the truck by 1.83m, at t = 0.

the box has a certain acceleration... the rear of the truck has a certain accleration... at what time does the rear of the truck catch up to the box...
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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