Problem 31 - University Physics with Modern 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 exerted on the chair is 93.5 N.

Step 1: Identify Forces Acting on the Chair

Start by identifying all the forces acting on the chair. These include the gravitational force exerted downward by the chair's weight, the applied force of 40.0 N acting at a 37.0-degree angle below horizontal, the normal force exerted by the floor upwards, and the frictional force acting opposite to the direction of motion.

Step 2: Draw the Free-Body Diagram

Sketch the chair and include vectors representing each force. Draw the weight (mg) vertically downward, the applied force (F) at a 37° angle to the horizontal pointing into the floor, the normal force (N) vertically upward from the chair, and friction (f) horizontally opposite to the direction of motion. Use labeled arrows to clearly identify each force.

Step 3: Resolve the Applied Force into Components

Resolve the force \(F = 40.0 \, \mathrm{N}\) into horizontal and vertical components using trigonometry. The horizontal component is \(F_x = F \cdot \cos(37°)\) and the vertical component is \(F_y = F \cdot \sin(37°)\).

Step 4: Write the Force Balance Equations

Consider Newton's laws for vertical and horizontal forces. Vertically, the sum of forces gives \(N + F_y = mg\). Horizontally, \(F_x - f = ma\), but we use the vertical forces to find the normal force first. Here, \(a\) is the acceleration and \(f\) is friction.

Step 5: Calculate the Vertical Force Components

Calculate the vertical component of the applied force: \(F_y = 40.0 \times \sin(37°)\). Calculate the gravitational force: \(mg = 12.0 \times 9.8\).

Step 6: Solve for Normal Force

Substitute the calculated values into the vertical force equation: \(N + F_y = mg\). Rearrange the equation to solve for the normal force, \(N = mg - F_y\).

Step 7: Perform Calculations

Plug in the numbers: \(F_y = 24.1 \, \mathrm{N}\) and \(mg = 117.6 \, \mathrm{N}\). Calculate \(N = 117.6 \, \mathrm{N} - 24.1 \, \mathrm{N} = 93.5 \, \mathrm{N}\).

Key Concepts

Free-Body Diagram

Understanding a free-body diagram is essential when analyzing the forces acting on an object. A free-body diagram is a visual representation used to depict all the forces acting on a single point or body. When you draw the diagram for the chair in the problem, you start by sketching the chair as a simple box or circle.
The forces should be represented by arrows pointing in the direction that the forces act. Here are the forces acting on the chair:

• The gravitational force, or weight, acting vertically downward, is represented by an arrow pointing towards the ground and is labeled with its magnitude, which is the weight of the chair (\(mg = 117.6 \mathrm{\, N}\)).
• The force applied by you, which is \(40.0\, \mathrm{N}\), is directed at an angle \(37.0^{\circ}\) below the horizontal. This force should be represented by a diagonal arrow pointing down and into the floor.
• The normal force, which is exerted by the floor, acts vertically upward against the chair. This arrow originates from the chair pointing vertically up and is typically labeled as \(N\).
• The frictional force is the resistance force exerted by the floor. It acts horizontally opposite to the chair's sliding motion and is labeled with an arrow in that direction, often marked \(f\).

Each of these arrows should be clearly labeled to understand better each force's role in the problem. This diagram allows you to visually interpret how the forces balance out and apply Newton's laws effectively.

Normal Force

The normal force is a crucial concept in physics and is fundamental when solving problems involving objects in contact with a surface. It is the force exerted by a surface perpendicular to the object resting on it. In the chair problem, the normal force is what the floor exerts on the chair to balance out other vertical forces, such as its weight and any additional vertical interaction.To calculate the normal force on the chair, you must consider both the chair's weight and the vertical component of the applied force. According to Newton's laws, the net vertical force must be zero for the chair to remain in constant contact with the floor without accelerating vertically. Thus, the sum of upward forces must equal the sum of downward forces.Here's how you apply Newton's second law to write the force balance equation:

• The gravitational force pulling the chair downward is \(mg = 12.0 \times 9.8 = 117.6 \mathrm{\, N}\).
• Resolve the applied force into its vertical component using trigonometry: \(F_y = 40.0 \times \sin(37^{\circ}) = 24.1 \mathrm{\, N}\).
• The equation for vertical forces is thus: \(N + F_y = mg\).

To find the normal force, rearrange this equation: \(N = mg - F_y\). Substitute the known values to find: \(N = 117.6 \mathrm{\, N} - 24.1 \mathrm{\, N} = 93.5 \mathrm{\, N}\). This is the normal force exerted by the floor on the chair.

Frictional Forces

Frictional forces always play a significant role in scenarios where objects slide across surfaces. They act to resist the movement of objects, and they originate from the contact between surfaces. In physics, understanding friction helps us explain why objects slow down or stop when there's no apparent cause.In our chair problem, since the floor is not frictionless, frictional forces oppose the chair's motion as it slides. The direction of the frictional force is always opposite to the direction of motion, acting horizontally along the floor.Here’s what you need to know about frictional forces:

• The magnitude of the frictional force can be described by \(f = \mu \cdot N\), where \(\mu\) is the coefficient of friction, and \(N\) is the normal force.
• Because the problem doesn't specify the value of \(\mu\), it's often calculated or given in different parts of questions. However, you can apply the concept to understand how strong friction would need to be to prevent sliding entirely.
• Keep in mind that the actual frictional force could be less if the chair is moving at a constant speed, meaning \(F_x = f\) for balance.

An understanding of this concept helps explain why more force is needed when pushing the chair faster or on a rougher surface compared to a smoother one, shedding light on everyday experiences tied to friction.