Problem 29 - University Physics with Modern Physics

Question

A chair of mass 12.0 kg is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force \(F =\) 40.0 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 is approximately 141.65 N.

Step 1: Understand the Problem

We need to determine the normal force exerted by the floor on the chair when a force is applied at an angle below the horizontal. Begin by analyzing the given data: the mass of the chair is 12.0 kg, and a force of 40.0 N is applied at an angle of 37.0° below the horizontal.

Step 2: Draw the Free-Body Diagram

Draw a free-body diagram representing all the forces acting on the chair. Include:- The gravitational force (\( F_g = mg \)) acting downward.- The normal force (\( N \)) acting upward.- The applied force (\( F \)) of 40.0 N at an angle of 37.0° below the horizontal.- The frictional force acting opposite to the direction of motion.

Step 3: Resolve the Applied Force

Resolve the applied force into its horizontal and vertical components:\[F_{x} = F \cdot \cos(37.0^{\circ})\quad \text{and} \quad F_{y} = F \cdot \sin(37.0^{\circ})\]where \( F = 40.0 \) N.

Step 4: Calculate the Vertical Force Components

Find the forces in the vertical direction:- The gravitational force \( F_g = mg = 12.0 \times 9.8 = 117.6 \, \text{N} \)- The vertical component of the applied force is \( F_y = 40.0 \times \sin(37.0^{\circ}) \approx 24.05 \, \text{N} \)- The normal force (\( N \)) must balance the other vertical forces.

Step 5: Apply Newton's Second Law in the Vertical Direction

Using Newton's second law for the vertical direction:\[N - F_y - mg = 0\]Solve for \( N \):\[N = F_y + mg = 24.05 + 117.6 \approx 141.65 \, \text{N}\]

Step 6: Conclusion

The normal force exerted by the floor on the chair is approximately \( 141.65 \) N.

Key Concepts

Free-Body Diagram

To understand how all forces affect the movement of an object, drawing a free-body diagram is crucial. This diagram is like a map of all the forces acting on an object, represented by arrows. Each arrow shows both the direction and magnitude of a force. For our chair example, imagine these forces:

• The gravitational force or weight, which is the force pulling the chair down toward the Earth. This can be described as \( F_g = mg \), where \( m \) is mass and \( g \) is the acceleration due to gravity.
• The normal force, usually acting perpendicular (at a right angle) to the surface. It keeps the chair from falling through the floor.
• The force you apply to the chair, pushing it at an angle. We can divide this force into two parts: a horizontal component pushing the chair across the floor and a vertical component pressing it down.
• The frictional force, which acts opposite to the direction of the applied force, resisting the chair's motion.

By putting all these forces on a single diagram, you gain a visual representation that helps solve problems using Newton's laws.

Normal Force

The normal force is a contact force that prevents objects from "falling" into the surface they rest on. It acts perpendicular to the surface. In our problem, the normal force occurs between the chair and the floor. It's interesting because while some forces like gravity are always present, normal force depends on the orientation and other applied forces.

When calculating the normal force, note that it's not just equal to the weight of the object. In our exercise, because the force is applied at an angle, the vertical component of the force affects the normal force. Thus, the normal force \( N \) balances the downward force from both the gravitational force and the vertical component of the applied force. For this chair:
- Gravity pulls the chair down: \( F_g = mg \)- The additional vertical force \( F_y = F \cdot \sin(37.0^{\circ}) \) also pushes it down.
Using Newton's second law, we find the normal force by summing these vertical forces: \( N = F_y + mg \). This gives the chair the overall vertical stability needed to counterbalance the downward forces.

Force Components

In physics, when a force is applied at an angle, it's useful to break it into components—horizontal and vertical—using trigonometry. This allows us to better estimate how the force influences motion in each direction. For a force \( F \) applied at an angle, we can split it into:

• Horizontal Component (\( F_x \)): This part pushes the object sideways. It's calculated using the cosine of the angle: \( F_x = F \cdot \cos(\theta) \).
• Vertical Component (\( F_y \)): This part affects how much the object is pressed against the ground. It's calculated with the sine of the angle: \( F_y = F \cdot \sin(\theta) \).

In the exercise, you push the chair with a force of 40.0 N at 37.0° below the horizontal. By resolving this force into components:- The horizontal component \( F_x = 40.0 \cdot \cos(37.0^{\circ}) \) provides the sideways movement.- The vertical component \( F_y = 40.0 \cdot \sin(37.0^{\circ}) \) adds to the downward force on the chair.
By understanding these components, deciphering the exact effects of forces applied at different angles becomes more manageable, providing clarity for force interactions described by Newton's laws.