Problem 41 - College Physics

Question

A chair of mass 12.0 \(\mathrm{kg}\) is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force \(F=40.0 \mathrm{N}\) that is directed at an angle of \(37.0^{\circ}\) below the horizontal, and the chair slides along the floor. (a) Draw a clearly labeled free-body diagram for the chair. (b) Use your diagram and Newton's laws to calculate the normal force that the floor exerts on the chair.

Step-by-Step Solution

Answer

The normal force the floor exerts on the chair is 93.6 N.

Step 1: Draw the Free-Body Diagram

Start by sketching the chair. Identify all the forces acting on the chair. The forces include:1. Gravitational force \(F_g = mg\) acting downward, where \(m = 12.0\, \text{kg}\) and \(g = 9.8\, \text{m/s}^2\).2. Applied force \(F = 40.0\, \text{N}\) acting at an angle \(37^{\circ}\) below the horizontal.3. Normal force \(N\), which acts perpendicular and upward from the floor.4. Frictional force \(f\), which acts opposite to the direction of movement.Ensure that these vectors are clearly labeled and point in the correct directions on your diagram.

Step 2: Calculate the Horizontal and Vertical Components of the Applied Force

To find the impact of the applied force on the normal force, decompose it into horizontal and vertical components.The horizontal component is given by:\[ F_{x} = F \cos(37^{\circ}) \]The vertical component is given by:\[ F_{y} = F \sin(37^{\circ}) \]Calculate these components using the given force \( F = 40.0 \text{N} \). Substituting the values, \[ F_{x} = 40.0 \cos(37^{\circ}) \approx 31.98\, \text{N} \]\[ F_{y} = 40.0 \sin(37^{\circ}) \approx 24.0\, \text{N} \]

Step 3: Apply Newton's Second Law in the Vertical Direction

Using Newton's second law, set up the equilibrium in the vertical direction as the chair is not moving vertically:\[ N + F_{y} = mg \]Rearrange to solve for the normal force \(N\):\[ N = mg - F_{y} \]Substitute the known values:\[ N = (12.0 \times 9.8) - 24.0 \]Calculate to find:\[ N = 117.6 - 24.0 = 93.6 \text{ N} \]

Step 4: Final Calculation and Conclusion

Ensure calculations are correct and interpret the result:- The normal force \( N \) is the force exerted by the floor that supports the chair's weight. Through previously calculated values, the correct normal force is \( N = 93.6 \text{ N}\).- Check that this value of \(N\) makes sense considering that the applied force has a downward vertical component that subtracts from the gravitational force.

Key Concepts

Understanding Free-Body Diagrams

A free-body diagram is essential for visualizing the forces acting on an object, like our chair. It's a simple sketch that shows all forces with vectors. Here's how you approach it:

• Identify the Object: In our problem, the object is a chair.
• List the Forces:
  • The gravitational force acts downward, which is the chair's weight, calculated as \( F_g = mg \), where \( m = 12.0\, \text{kg} \) and \( g = 9.8\, \text{m/s}^2 \).
  • The applied force of \( 40.0\, \text{N} \) makes an angle of \( 37^{\circ} \) with the horizontal.
  • The normal force is a supportive force from the ground, acting perpendicular and upward to counteract the gravitational force.
  • The frictional force opposes the chair's movement on the floor.

Mark these vectors on the diagram, keeping in mind their direction relative to the object. This visual aid simplifies analyzing the problem, helping you apply Newton's Laws efficiently.

Calculating the Normal Force

The normal force in this context is the force exerted by a surface to support the weight of an object on it, acting perpendicular to the surface. Calculating it involves understanding the components of the forces at play.To find the normal force, you start by using Newton's Laws, which state that forces in equilibrium equal zero in the non-moving direction. Here, our focus is the vertical balance:

• Equation of Vertical Forces: The sum of the upward force (normal force) and the vertical component of the applied force must equal the gravitational force: \( N + F_{y} = mg \).
• Solving for the Normal Force: Rearrange the equation to find \( N \): \( N = mg - F_{y} \).
• Substitute Known Values: Plug in \( mg = 117.6\, \text{N} \) (from the gravitational force calculation) and \( F_{y} \approx 24.0 \text{N} \): \( N = 117.6 - 24.0 = 93.6\, \text{N} \).

This result shows how the normal force is reduced by the downward component of the applied force. A clear understanding of this process is crucial when analyzing similar force scenarios.

Breaking Down Force Components

To understand how forces affect a system, like the chair in this problem, it's critical to separate them into horizontal and vertical components. This makes it easier to analyze their individual effects.

• Applied Force Components: Our given force \( F = 40.0 \text{N} \) is angled below the horizontal plane. To find its components:
  • The horizontal component: \( F_{x} = F \cos(37^{\circ}) \).
  • The vertical component: \( F_{y} = F \sin(37^{\circ}) \).
• Calculate with Angles: Use trigonometry to find \( F_{x} \approx 31.98\, \text{N} \) and \( F_{y} \approx 24.0\, \text{N} \).

Recognizing how a single angled force splits into these parts is key. It simplifies using Newton's Laws to analyze and solve force-related problems, like calculating the normal force. This approach can be applied to a multitude of scenarios involving forces at various angles.