A box of books weighing 290 N is shoved across the floor of an apartment by a force of 400 N exerted downward at an angle of 35.0° below the horizontal. If the coefficient of kinetic friction between box and floor is 0.57, how long does it take to move the box 3.80 m, starting from rest?

Calculate the acceleration rate, a, using Newton's second law:

Fnet = M a

Then get the required time t using

3.80 m = (a/2) t^2

The net force Fnet is the difference between 400 N and the friction force.

To find the time it takes to move the box, we can follow these steps:

1. Determine the net force acting on the box:
- The force exerted downward at an angle of 35.0° below the horizontal can be split into horizontal and vertical components.
- The vertical component is `400 N * sin(35.0°)`
- The horizontal component is `400 N * cos(35.0°)`
- The weight of the box (force due to gravity) is `290 N` acting vertically downward.
- The normal force exerted by the floor is equal in magnitude and opposite in direction to the weight of the box, so it is also `290 N` acting vertically upward.
- The frictional force acting on the box can be found using the coefficient of kinetic friction: `frictional force = coefficient of kinetic friction * normal force`

2. Calculate the net force:
- The net force acting on the box is the horizontal component of the applied force minus the frictional force. Since they are in opposite directions, we subtract them: `net force = horizontal component of applied force - frictional force`

3. Apply Newton's second law of motion:
- Newton's second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration: `net force = mass * acceleration`
- Rearranging the equation to solve for acceleration: `acceleration = net force / mass`

4. Determine the acceleration:
- To find the acceleration, we divide the net force by the mass of the box. Since we are given the weight of the box, we can calculate its mass using the formula: `mass = weight / acceleration due to gravity`.
- The acceleration due to gravity is approximately `9.8 m/s²`.
- Therefore, `mass = 290 N / 9.8 m/s²`

5. Calculate the time taken to move 3.80 m:
- We can use the equations of motion to find the time required to move a distance.
- The equation to find the final velocity of an object is: `final velocity² = initial velocity² + 2 * acceleration * distance`
- Since the box starts from rest, the initial velocity is `0 m/s`.
- Rearranging the equation to solve for time: `time = (final velocity - initial velocity) / acceleration`
- Plugging in the values, `final velocity² = 0 + 2 * acceleration * 3.80 m`

6. Substitute the values and solve for time:
- Substitute the calculated values into the equation: `time = (sqrt(2 * acceleration * distance) - 0) / acceleration`

By following these steps and substituting the given values into the equations, you can find the time it takes to move the box.