A chair of weight 100N lies atop a horizontal floor; the floor is not frictionless. You push on the chair with a force of F = 43.0N directed at an angle of 35.0^\circ below the horizontal and the chair slides along the floor.

Using Newton's laws, calculate n, the magnitude of the normal force that the floor exerts on the chair.
Express your answer in newtons.

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To calculate the magnitude of the normal force (n) that the floor exerts on the chair, we need to consider the forces acting on the chair.

First, let's resolve the applied force into its horizontal and vertical components. The horizontal component (F_x) can be calculated using the formula:

F_x = F * cos(theta)

where F is the magnitude of the applied force (43.0N) and theta is the angle below the horizontal (35.0°):

F_x = 43.0N * cos(35.0°)

Next, we need to determine the frictional force (F_friction) acting on the chair. It can be calculated using the formula:

F_friction = μ * n

where μ is the coefficient of friction between the chair and the floor, and n is the magnitude of the normal force. Since the chair is sliding along the floor, the frictional force should be equal to the applied horizontal force (F_x):

F_friction = F_x

Now, we can solve for n (the normal force) by rearranging the equation:

n = F_friction / μ

Substituting in the known values:

n = (43.0N * cos(35.0°)) / μ

Since the coefficient of friction is not given in the question, it is not possible to determine the exact magnitude of the normal force (n) without this information. The normal force will depend on the friction coefficient (μ) between the chair and the floor.

To calculate the normal force exerted by the floor on the chair, we need to consider the forces acting on the chair in the vertical direction.

1. Determine the vertical component of the applied force:
The force you applied on the chair has an angle of 35.0 degrees below the horizontal. We can calculate the vertical component of this force by multiplying the magnitude of the force (43.0N) by the sine of the angle (35.0 degrees):
F_vert = F * sin(alpha) = 43.0N * sin(35.0 degrees) = 24.44N

2. Calculate the gravitational force:
The weight of the chair is given as 100N. The force due to gravity always acts straight down, so its vertical component is equal to its magnitude:
F_grav = 100N

3. Determine the net force in the vertical direction:
Since the chair is sliding along the floor horizontally, we can assume that there is no vertical acceleration. Therefore, the net force in the vertical direction must be zero:
F_net = F_vert + F_grav + F_normal = 0

4. Solve for the normal force:
Rearranging the equation for F_net, we find:
F_normal = - F_vert - F_grav = - 24.44N - 100N = - 124.44N

The negative sign indicates that the normal force is directed upwards, opposing the force due to gravity.

Finally, we take the magnitude of the normal force, as stated in the question:
n = |F_normal| = |-124.44N| = 124.44N.

Therefore, the magnitude of the normal force that the floor exerts on the chair is 124.44 newtons.