Use the steps outlined above to find the magnitude of the acceleration $ a $ of a chair and the magnitude of the normal force $ F_N $ acting on the chair:

Yusef pushes a chair of mass $ m = 50.0 $ kg across a carpeted floor with a force vector $ F_p $ of magnitude $ F_p = 144 $ N directed at $ \theta = 35.0 $ degrees below the horizontal (Figure 1). The magnitude of the kinetic frictional force between the carpet and the chair is $ F_k = 97.4 $ N.

1. What is the magnitude of the acceleration $ a $ of the chair?
2. What is the magnitude of the normal force $ F_N $ acting on the chair?

Express your answers, separated by a comma, in meters per second squared and newtons to three significant figures.

$ a, F_N = $ m/s², N

To find the magnitude of the acceleration a of the chair, we need to calculate the net force acting on the chair. The net force is the difference between the applied force and the frictional force.

First, let's calculate the horizontal and vertical components of the applied force Fₚ. The horizontal component is given by Fₚ * cos(θ) and the vertical component is given by Fₚ * sin(θ).

The horizontal component of the applied force is Fₚ * cos(35.0°) = 144 N * cos(35.0°) = 118.4 N.

The vertical component of the applied force is Fₚ * sin(35.0°) = 144 N * sin(35.0°) = 82.6 N.

Next, let's calculate the frictional force Fₖ. The magnitude of the kinetic frictional force is given as Fₖ = 97.4 N.

Since the chair is moving horizontally, the frictional force acts in the opposite direction to the applied horizontal force. Therefore, the frictional force is -97.4 N.

Now, let's calculate the net horizontal force acting on the chair. The net horizontal force is the sum of the applied horizontal force and the frictional force.

Net horizontal force = Applied horizontal force + Frictional force = 118.4 N + (-97.4 N) = 21 N.

Finally, we can calculate the acceleration of the chair using Newton's second law of motion, F = m * a. Rearranging the equation, we have a = F / m.

Acceleration a = Net horizontal force / mass = 21 N / 50.0 kg = 0.42 m/s².

To find the magnitude of the normal force Fɴ acting on the chair, we can use the equation Fₖ = μₖ * Fₙ, where Fₖ is the kinetic frictional force, μₖ is the coefficient of kinetic friction, and Fₙ is the normal force.

Since the chair is on a horizontal surface, the normal force is equal in magnitude to the weight of the chair, which is given by the equation Fₙ = m * g, where m is the mass of the chair and g is the acceleration due to gravity.

Normal force Fɴ = Weight of the chair = m * g = 50.0 kg * 9.8 m/s² = 490 N.

Therefore, the magnitude of the acceleration a of the chair is 0.42 m/s² and the magnitude of the normal force Fɴ acting on the chair is 490 N.

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