A box of books weighing 229 N is shoved across the floor by a force of 420 N exerted downward at an angle of 35° below the horizontal.If µk between the box and the floor is 0.57, how long does it take to move the box 9 m, starting from rest? (If the box will not move, enter 0.)

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To find the time it takes to move the box, we can use Newton's second law of motion - specifically, the equation for the net force:

Net force = mass × acceleration

In this case, the net force is equal to the force applied minus the force of friction. The force applied is 420 N downward at an angle of 35° below the horizontal. The force of friction can be calculated using the formula:

Force of friction = coefficient of kinetic friction (µk) × normal force

The normal force is equal to the weight of the box, which is the force of gravity acting on it. Hence, the normal force is equal to its mass multiplied by acceleration due to gravity (9.8 m/s^2).

Using this information, let's calculate the force of friction:

Force of friction = 0.57 × (mass × 9.8)

We are given the weight of the box, which is 229 N. Since weight is equal to mass times acceleration due to gravity, we can rearrange the equation to solve for the mass:

mass = weight / acceleration due to gravity

mass = 229 N / 9.8 m/s^2

Substituting this value into the equation for the force of friction:

Force of friction = 0.57 × (229 N / 9.8 m/s^2 × 9.8 m/s^2)

Now, we can calculate the net force:

Net force = 420 N - Force of friction

Next, we can use the net force equation to find the acceleration:

Net force = mass × acceleration

Rearranging the equation:

Acceleration = Net force / mass

Substituting the values we found:

Acceleration = Net force / (229 N / 9.8 m/s^2)

Now, we have all the required information to calculate the time it takes to move the box a distance of 9 m. Since it starts from rest, we can use the following kinematic equation:

Distance = (initial velocity × time) + (1/2 × acceleration × time^2)

Since the initial velocity is zero, the equation simplifies to:

Distance = (1/2 × acceleration × time^2)

Substituting the given values:

9 m = (1/2 × acceleration × time^2)

Now, we can solve for time:

time^2 = (2 × distance) / acceleration

Taking the square root:

time = √(2 × distance / acceleration)

Plugging in the values:

time = √(2 × 9 m / acceleration)

Finally, calculate the value of time.