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Chapter 4: Problem 7

The air exerts a forward force of \(10 \mathrm{~N}\) on the propeller of a \(0.20-\mathrm{kg}\) model airplane. If the plane accelerates forward at \(2.0 \mathrm{~m} / \mathrm{s}^{2}\), what is the magnitude of the resistive force exerted by the air on the airplane?

Short Answer

Expert verified
The magnitude of the resistive force exerted by the air on the airplane is \(9.6 \) N.

Step by step solution

01

Understand the Forces at Play

First, realize that the net force acting on the airplane is equal to the difference between the forward force exerted by the propeller and the resistive force exerted by the wind.
02

Write Newton's Second Law

According to Newton's second law, the net force acting on an object is equal to its mass times its acceleration. Mathematically, this is expressed as \( F_{net} = ma \). Here, \( F_{net} \) is the net force, m is the mass, and a is the acceleration. In this case, the net force is also equal to the forward force (F) minus the resistive force (Fr). So, the equation becomes \( F - Fr = ma \).
03

Substitute the given quantities into the equation

The propeller force (F) is given as 10 N, the mass (m) of the plane is 0.20 kg, and the acceleration (a) is 2 m/s虏. Substituting these values gives us \( 10 - Fr = 0.20 * 2 \).
04

Solve for the Resistive Force

Solve the equation for Fr. Simplify the right side first to get \( 10 - Fr = 0.4 \). Then, add Fr to both sides, and subtract 0.4 from both sides to isolate Fr, giving the equation \( Fr = 10 - 0.4 = 9.6 \) N. This means the resistive force exerted by the air on the airplane is 9.6 N.

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Key Concepts

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

Forces
Forces are pushes or pulls that can cause an object to move, stop, or change direction. In the context of the model airplane, we deal with two important forces: the forward force exerted by the propeller and the resistive force exerted by the air. The forward force propels the plane forward, while the resistive force acts in the opposite direction, opposing the motion. Understanding these forces helps us comprehend how objects move and respond to external influences.

Key points about forces include:
  • Forces have both magnitude and direction, making them vector quantities.
  • The net force determines the change in motion of an object.
  • Opposing forces like friction or air resistance can slow down or halt the motion of an object.

In our exercise, recognizing the two opposing forces allows us to calculate the net force and subsequently understand how the airplane accelerates.
Acceleration
Acceleration describes how quickly an object's velocity changes with time. It is a crucial aspect of Newton鈥檚 Second Law, which connects forces and motion. Generally, when a net force acts on an object, it leads to acceleration. In the exercise given, the model airplane accelerates at 2 m/s虏, indicating an increase in velocity in the direction of the net force.

Further breakdown of acceleration involves:
  • It's a vector quantity, meaning it includes both size and direction.
  • Acceleration can result from a change in speed or direction (or both).
  • The formula to determine acceleration is: \[ a = \frac{F_{net}}{m} \] where \( a \) is the acceleration, \( F_{net} \) is the net force, and \( m \) is the mass of the object.

In our scenario, understanding acceleration helps us find the relationship between the forces at work and how they affect the movement of the airplane.
Mass
Mass is a measure of the amount of matter in an object and plays a significant role in inertia and the response of objects to forces. In the context of Newton鈥檚 Second Law, mass is pivotal in determining how much an object will accelerate when acted upon by a force. A more massive object will require a larger force to achieve the same acceleration as a less massive object.

Important points about mass include:
  • Mass is a scalar quantity, meaning it has magnitude but no direction.
  • It is measured in kilograms (kg) in the metric system.
  • The formula for Newton's Second Law is \[ F_{net} = ma \] where \( F_{net} \) is the net force, \( m \) is the mass, and \( a \) is the acceleration.

In the exercise, with a mass of 0.20 kg, the model airplane's acceleration helps determine the effect of the net force after accounting for resistance. This shows how critical mass is in calculating the dynamics of motion.

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Most popular questions from this chapter

A high diver of mass \(70.0 \mathrm{~kg}\) jumps off a board \(10.0 \mathrm{~m}\) above the water. If her downward motion is stopped \(2.00 \mathrm{~s}\) after she enters the water, what average upward force did the water exert on her?

A crate of mass \(45.0 \mathrm{~kg}\) is being transported on the flatbed of a pickup truck. The coefficient of static friction between the crate and the truck's flatbed is \(0.350\), and the coefficient of kinetic friction is \(0.320 .\) (a) The truck accelerates forward on level ground. What is the maximum acceleration the truck can have so that the crate does not slide relative to the truck's flatbed? (b) The truck barely exceeds this acceleration and then moves with constant acceleration, with the crate sliding along its bed. What is the acceleration of the crate relative to the ground?

A \(3.00-\mathrm{kg}\) block starts from rest at the top of a \(30.0^{\circ}\) incline and slides \(2.00 \mathrm{~m}\) down the incline in \(1.50 \mathrm{~s}\). Find (a) the acceleration of the block, (b) the coefficient of kinetic friction between the block and the incline, (c) the frictional force acting on the block, and (d) the speed of the block after it has slid \(2.00 \mathrm{~m}\).

The force exerted by the wind on the sails of a sailboat is 390 Nnorth. The water exerts a force of 180 Neast. If the boat (including its crew) has a mass of \(270 \mathrm{~kg}\), what are the magnitude and direction of its acceleration?

Consider a solid metal sphere (S) a few centimeLers in diameter and a feather (F). For each quantity in the list that follows, indicate whether the quantity is the same, greater, or lesser in the case of \(\mathrm{S}\) or in that of \(\mathrm{F}\). Explain in each case why you gave the answer you did. Here is the list: (a) the gravitational force, (b) the time it will take to fall a given distance in air, (c) the time it will take to fall a given distance in vacuum, (d) the total force on the object when falling in vacuum.
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