A box of books weighing 337 N is shoved across the floor by a force of 467 N exerted downward at an angle of 35° below the horizontal.

(a) If μk between the box and the floor is 0.56, how long does it take to move the box 4.00 m starting from rest?
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(b) What is the maximum coefficient of kinetic friction between the box and the floor that allows the box to move from this applied force.

To solve these problems, we can use Newton's second law of motion and the equations of motion.

(a) First, let's find the net force acting on the box. The net force is the horizontal component of the applied force minus the force of kinetic friction. Using trigonometry, we find the horizontal component of the applied force:

F_horizontal = 467 N * cos(35°)
F_horizontal ≈ 467 N * 0.819 = 382.49 N

Next, let's calculate the force of kinetic friction:

F_friction = μk * N
where N is the normal force. In this case, the normal force is equal to the weight of the box:

N = mg
N = 337 N

F_friction = μk * 337 N
F_friction = 0.56 * 337 N ≈ 188.72 N

Now we can find the net force:

Net force = F_horizontal - F_friction
Net force = 382.49 N - 188.72 N
Net force = 193.77 N

To find the acceleration of the box, we can use Newton's second law of motion:

F_net = ma
193.77 N = m * a

Next, we can use the equation of motion to find the time it takes to move the box a distance of 4.00 m starting from rest:

s = ut + 0.5at^2
4.00 m = 0.5 * a * t^2
8.00 m = a * t^2

Finally, we have two equations with two unknowns (m and t). We can solve them simultaneously to find the time it takes to move the box:

193.77 N = m * a
8.00 m = a * t^2

Solving these equations will give us the values of m and t.

(b) To find the maximum coefficient of kinetic friction that allows the box to move, we need to find the net force when the box is just about to start moving. At the point of impending motion, the force of kinetic friction will be at its maximum.

Using the same method as above, we can find the force of kinetic friction when the box is just about to start moving. We'll assume that the net force is equal to zero at this point:

Net force = F_horizontal - F_friction
0 = F_horizontal - F_friction

We can rearrange this equation to solve for the maximum coefficient of kinetic friction:

F_friction = F_horizontal
μk * N = F_horizontal

Plugging in the known values, we can solve for the maximum coefficient of kinetic friction.