A chair of mass 14.0kg is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force F = 40.0N that is directed at an angle of 43.0∘ below the horizontal and the chair slides along the floor.

To solve this problem, we can break down the force into its horizontal and vertical components.

Given:
Mass of the chair (m) = 14.0 kg
Force applied (F) = 40.0 N
Angle (θ) = 43.0°

Step 1: Find the vertical force component.
The vertical component (Fv) can be determined by multiplying the force (F) with the sine of the angle (θ).
Fv = F * sin(θ)
Fv = 40.0 N * sin(43.0°)

Step 2: Find the horizontal force component.
The horizontal component (Fh) can be determined by multiplying the force (F) with the cosine of the angle (θ).
Fh = F * cos(θ)
Fh = 40.0 N * cos(43.0°)

Step 3: Calculate the force of friction (ff).
The force of friction is equal to the coefficient of friction (μ) multiplied by the normal force (Fn), which can be determined using the vertical force component (Fv).
ff = μ * Fn
Fn = m * g
ff = μ * (m * g)

Note: The normal force (Fn) opposes the gravitational force acting on the chair.

Step 4: Determine the acceleration (a) of the chair.
The net force acting on the chair (Fnet) is equal to the horizontal force component (Fh) minus the force of friction (ff).
Fnet = Fh - ff
ma = Fnet
a = Fnet / m

Step 5: Calculate the distance (d) traveled by the chair.
The distance traveled (d) by the chair can be determined using the equation of motion:
d = (1/2) * a * t^2

By solving these equations, we can find the vertical and horizontal components of the force, the force of friction, the acceleration, and the distance traveled by the chair.

To determine the acceleration of the chair, we need to analyze the forces acting on it. In this case, there are three forces that we should consider: the force of gravity, the normal force exerted by the floor, and the force you applied.

Let's break down these forces:

1. Force of gravity (weight): The chair has a mass of 14.0 kg, so its weight can be calculated using the equation: weight = mass × acceleration due to gravity. Assuming the acceleration due to gravity is 9.8 m/s^2, the weight of the chair is: weight = 14.0 kg × 9.8 m/s^2 = 137.2 N. This force is acting vertically downward.

2. Normal force: The normal force is the force exerted by a surface to support the weight of an object resting on it. It acts perpendicular to the surface. In this case, the chair is on a horizontal floor, so the normal force would be equal in magnitude and opposite in direction to the weight of the chair. Therefore, the normal force is also 137.2 N.

3. Applied force: You push the chair with a force of 40.0 N, directed at an angle of 43.0 degrees below the horizontal. To determine the horizontal and vertical components of this force, we can use trigonometry. The horizontal component is F_horizontal = F × cos(angle) = 40.0 N × cos(43.0°) = 29.0 N. The vertical component is F_vertical = F × sin(angle) = 40.0 N × sin(43.0°) = 25.9 N.

Now that we have analyzed the forces, we can determine the net force acting on the chair in the horizontal direction. Since there is no horizontal acceleration, the net force is zero. Therefore, the force you applied in the horizontal direction (29.0 N) must be equal in magnitude and opposite in direction to the force of friction.

Finally, we can calculate the coefficient of friction between the chair and the floor using the equation: coefficient of friction = force of friction / normal force. In this case, the force of friction is equal to 29.0 N, and the normal force is 137.2 N. Thus, the coefficient of friction is: coefficient of friction = 29.0 N / 137.2 N ≈ 0.211.

It's important to note that the acceleration of the chair will depend on the coefficient of friction and the net force acting in the horizontal direction. But, since the net force is zero, the chair will not accelerate horizontally and will remain at rest or move at a constant velocity.

M*g = 14 * 9.8 = 137.2 N. = Wt. of chair.

Fn = Mg + F*sin43 = 137.2 + 40*sin43 =
164.5 N. = Normal force.

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