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 35.1° 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 4.20 m, starting from rest?

To find the time it takes to move the box, we need to use the equation of motion. The equation we will use is:

\[s = ut + \frac{1}{2}at^2\]

Where:
- \(s\) is the distance moved by the box (4.20 m),
- \(u\) is the initial velocity of the box (0 m/s as it starts from rest),
- \(a\) is the acceleration of the box,
- And \(t\) is the time taken to move the box.

We can rearrange the equation to solve for time:

\[t = \sqrt{\frac{2s}{a}}\]

To find the acceleration of the box, we can use the equation:

\[F_{net x} = ma\]

Where:
- \(F_{net x}\) is the net force acting on the box in the horizontal direction,
- \(m\) is the mass of the box, and
- \(a\) is the acceleration of the box.

The net force acting on the box in the horizontal direction is the horizontal component of the applied force minus the force of friction. The horizontal component of the force is given by:

\[F_{horizontal} = F \cdot \cos(\theta)\]

Where:
- \(F\) is the applied force (400 N),
- \(\theta\) is the angle below the horizontal (35.1°), and
- \(F_{horizontal}\) is the horizontal component of the applied force.

The force of friction can be calculated using:

\[f_{friction} = \mu \cdot f_{normal}\]

Where:
- \(\mu\) is the coefficient of kinetic friction (0.57),
- \(f_{normal}\) is the normal force acting on the box.

The normal force is equal in magnitude and opposite in direction to the weight of the box, which is given by:

\[f_{normal} = mg\]

Where:
- \(m\) is the mass of the box, and
- \(g\) is the acceleration due to gravity (9.8 m/s^2).

Since the weight of the box is equal to the force, we have:

\[f_{normal} = F\]

Now we can substitute the expressions for \(F_{horizontal}\) and \(f_{friction}\) to find the net force acting on the box:

\[F_{net x} = F_{horizontal} - f_{friction}\]

Substituting the values, we get:

\[F_{net x} = (400 \, N) \cdot \cos(35.1°) - (0.57) \cdot (290 \, N)\]

We can now solve for the acceleration of the box:

\[a = \frac{F_{net x}}{m}\]

Finally, we can substitute the values for distance (\(s\)), acceleration (\(a\)) into the equation for time:

\[t = \sqrt{\frac{2s}{a}}\]

To find the time it takes to move the box, we need to use Newton's second law of motion, which states that the net force (F_net) acting on an object is equal to its mass (m) multiplied by its acceleration (a): F_net = m * a.

First, we need to find the net force acting on the box. The net force is the vector sum of all the forces acting on the box. In this case, there are two forces: the force pushing down at an angle (400 N) and the force of friction opposing the motion.

Let's break down the force pushing down at an angle into its horizontal and vertical components. The horizontal component is given by F_horizontal = F_push * cos(angle), where F_push is the force pushing down (400 N) and angle is the angle it makes with the horizontal (35.1°). Therefore, F_horizontal = 400 N * cos(35.1°).

The vertical component does not affect the motion horizontally, so we will not consider it in this calculation.

Now let's find the force of friction. The force of friction (F_friction) is given by the coefficient of friction (μ) multiplied by the normal force (F_normal) acting on the box. The normal force is equal to the weight of the box, which is the mass (m) of the box multiplied by the acceleration due to gravity (g). F_normal = m * g, where g is approximately 9.8 m/s^2.

Given the weight of the box (290 N), we can find the mass by dividing the weight by the acceleration due to gravity: m = 290 N / 9.8 m/s^2.

Now we can calculate the force of friction: F_friction = μ * F_normal, where μ is the coefficient of kinetic friction (0.57).

Next, we can calculate the net force: F_net = F_horizontal - F_friction.

Since we know the net force acting on the box (F_net), we can rearrange Newton's second law of motion to solve for acceleration: a = F_net / m.

To find the time it takes to move the box a certain distance, we can use the equation: s = ut + (1/2)at^2, where s is the distance (4.20 m), u is the initial velocity (0 m/s), a is the acceleration, and t is the time.

Since the box starts from rest, the initial velocity (u) is zero. The equation simplifies to: s = (1/2)at^2.

Rearranging the equation to solve for time (t), we get: t = sqrt((2s) / a).

Now we have all the necessary information to calculate the time it takes to move the box. Let's plug in the values and calculate the answer:

1. Calculate F_horizontal: F_horizontal = 400 N * cos(35.1°).
2. Calculate the mass (m): m = 290 N / 9.8 m/s^2.
3. Calculate F_friction: F_friction = 0.57 * m * g.
4. Calculate F_net: F_net = F_horizontal - F_friction.
5. Calculate acceleration (a): a = F_net / m.
6. Calculate time (t): t = sqrt((2 * s) / a), where s is 4.20 m.

By following these steps, you should be able to find the time it takes to move the box 4.20 m starting from rest.