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Chapter 4: Problem 31

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 \(F=40.0 \mathrm{N}\) 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.

Short Answer

Expert verified
The normal force is approximately 93.72 N.

Step by step solution

01

Understand the Forces Acting on the Chair

In this situation, the chair experiences several forces. These include the force of gravity acting downwards (weight), the normal force from the floor acting upwards, the applied force at an angle, and the force of friction. For part (a), our task is to clearly identify and label these forces in a free-body diagram, which aids in visualizing their interactions.
02

Draw the Free-Body Diagram

To create a free-body diagram, depict the chair as a point. The forces acting are:- The gravitational force: downward arrow labeled \( mg = 12.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 117.72 \, \text{N} \).- The normal force: upward arrow labeled \( N \).- The applied force: arrow pointing at a 37掳 angle downward from the horizontal right, labeled \( F = 40.0 \, \text{N} \).- The force of friction opposing motion will be dependent on the normal force and is not needed for part (a), but is helpful to note.
03

Resolve the Applied Force into Components

Since the force \( F \) is at an angle, it has horizontal and vertical components. Calculate these components:- Horizontal component: \( F_x = 40.0 \, \text{N} \times \cos(37掳) \).- Vertical component: \( F_y = 40.0 \, \text{N} \times \sin(37掳) \).
04

Apply Newton's Second Law for Vertical Forces

In the vertical direction, the forces must balance because the chair does not accelerate vertically. The sum of forces is:\[ N + F_y - mg = 0 \]Solve for \( N \):\[ N = mg - F_y \]
05

Calculate the Normal Force

Insert the known values into the equation:- Find \( F_y \): \( F_y = 40.0 \, \text{N} \times \sin(37掳) = 24.0 \, \text{N} \).- Compute \( N \): \[ N = 117.72 \, \text{N} - 24.0 \, \text{N} = 93.72 \, \text{N} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Free-Body Diagram
A free-body diagram is a simple sketch that helps us visualize the forces acting on an object. In this exercise, the chair represents our object of interest. Here is how you construct the diagram:
  • **Gravity** acts downwards towards the center of Earth, which is represented by a downward arrow labeled with the force magnitude. For the chair, this force is the weight, given by \( mg = 12.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 117.72 \, \text{N} \).

  • The **normal force** acts perpendicular to the surface the object is resting on 鈥� in this case, it is the floor. This force is illustrated as an upward arrow labeled \( N \).

  • The **applied force** is shown as an arrow at a 37-degree angle below the horizontal to represent the direction of the push.
It's crucial to label each of these forces clearly, as they will be used in your calculations.
The Role of Normal Force
The normal force is the support force exerted by a surface to balance the weight of an object resting on it. Without it, objects would not be able to rest on surfaces.
For this scenario, the normal force is crucial. The chair is on a horizontal floor and is being pushed at an angle. You can calculate the normal force by considering all vertical forces.
According to Newton鈥檚 Second Law, the sum of forces in the vertical direction should equal zero, as there is no vertical acceleration. Thus:
\[ N + F_y - mg = 0 \]
Here, \( F_y \) is the vertical component of the applied force. Solving for the normal force gives:
\[ N = mg - F_y \]
This equation shows that the normal force not only depends on the gravitational force but also on how the applied force influences the vertical balance. By resolving the forces correctly, you ensure stability in your answer.
Breaking Down Force Components
Whenever a force is applied at an angle, it has to be broken into two components: horizontal and vertical. Knowing these components is essential for solving problems involving inclined or angled forces.
  • The **horizontal component** (\( F_x \)) can be found using the formula:
    \[ F_x = F \cdot \cos(\theta) \]
    For our chair, \( F_x = 40.0 \, \text{N} \times \cos(37掳) \).

  • The **vertical component** (\( F_y \)) is calculated as:
    \[ F_y = F \cdot \sin(\theta) \]
    In this example, \( F_y = 40.0 \, \text{N} \times \sin(37掳) = 24.0 \, \text{N} \).
By knowing these components, we understand how the force impacts the motion and how it affects the normal force and friction. This breakdown assists in analyzing the overall effect on the chair's movement.

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Most popular questions from this chapter

BIO Human Biomechanics. The fastest served tennis ball, served by "Big Bill" Tilden in \(1931,\) was measured at 73.14 \(\mathrm{m} / \mathrm{s} .\) The mass of a tennis ball is \(57 \mathrm{g},\) and the ball is typically in contact with the tennis racquet for \(30.0 \mathrm{ms},\) with the ball starting from rest. Assuming constant acceleration, (a) what force did Big Bill's racquet exert on the tennis ball if he hit it essentially horizontally? (b) Draw free-body diagrams of the tennis ball during the serve and just after it moved free of the racquet.

A spacecraft descends vertically near the surface of Planet X. An upward thrust of 25.0 \(\mathrm{kN}\) from its engines slows it down at a rate of \(1.20 \mathrm{m} / \mathrm{s}^{2},\) but it speeds up at a rate of 0.80 \(\mathrm{m} / \mathrm{s}^{2}\) with an upward thrust of 10.0 \(\mathrm{kN}\) (a) In each case, what is the direction of the acceleration of the spacecraft? (b) Draw a free- body diagram for the spacecraft. In each case, speeding up or slowing down, what is the direction of the net force on the spacecraft? (c) Apply Newton's second law to each case, slowing down or speeding up, and use this to find the spacecraft's weight near the surface of Planet X.

\(\mathrm{A}\) man is dragging a trunk up the loading ramp of a mover's truck. The ramp has a a slope angle of \(20.0^{\circ}\) , and the man pulls upward with a force \(\vec{F}\) whose direction makes an angle of \(30.0^{\circ}\) with the ramp (Fig. E4.4). (a) How large a force \(\vec{F}\) is necessary for the component \(F_{x}\) parallel to the ramp to be 60.0 \(\mathrm{N} ?\) (b) How large will the component \(F_{y}\) perpendicular to the ramp then be?

A ball is hanging from a long string that is tied to the ceiling of a train car traveling eastward on horizontal tracks. An observer inside the train car sees the ball hang motionless. Draw a clearly labeled free-body diagram for the ball if (a) the train has a uniform velocity, and (b) the train is speeding up uniformly. Is the net force on the ball zero in either case? Explain.

Two crates, one with mass 4.00 \(\mathrm{kg}\) and the other with mass \(6.00 \mathrm{kg},\) sit on the frictionless surface of a frozen pond, connected by a light rope (Fig. P4.43). A woman wearing golf shoes (so she can get traction on the ice) pulls horizontally on the 6.00 -kg crate with a force \(F\) that gives the crate an acceleration of 2.50 \(\mathrm{m} / \mathrm{s}^{2} .\) (a) What is the acceleration of the 4.00 -kg crate? (b) Draw a free-body diagram for the 4.00 -kg crate. Use that diagram and Newton's second law to find the tension \(T\) in the rope that connects the two crates. (c) Draw a free-body diagram for the 6.00 -kg crate. What is the direction of the net force on the 6.00 -kg crate? Which is larger in magnitude, force \(T\) or force \(F ?(\mathrm{d})\) Use part \((\mathrm{c})\) and Newton's second law to calculate the magnitude of the force \(F .\)
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