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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

An object of mass \(m\) is dropped from the roof of a building of height \(h .\) While the object is falling, a wind blowing parallel to the face of the building exerts a constant horizontal force \(F\) on the object. (a) How long does it take the object to strike the ground? Express the time \(t\) in terms of \(g\) and \(h\) (b) Find an expression in terms of \(m\) and \(F\) for the acceleration \(a_{x}\) of the object in the horizontal direction (taken as the positive \(x\) -direction). (c) How far is the object displaced horizontally before hitting the ground? Answer in terms of \(m, g, F_{t}\) and \(h\). \((d)\) Find the magnitude of the object's acceleration while it is falling, using the variables \(F, m\), and \(g\).

An \(80-\mathrm{kg}\) stuntman jumps from a window of a building situated \(30 \mathrm{~m}\) above a catching net. Assuming air resistance exerts a \(100-\mathrm{N}\) force on the stuntman as he falls, determine his velocity just before he hits the net.

A boat moves through the water with two forces acting on it. One is a \(2000-\mathrm{N}\) forward push by the water on the propeller, and the other is a \(1800-\mathrm{N}\) resistive force due to the water around the bow. (a) What is the acceleration of the \(1000-\mathrm{kg}\) boat? (b) If it starts from rest, how far will the boat move in \(10.0 \mathrm{~s}\) ? (c) What will its velocity be at the end of that time?

A 2 000-kg car is slowed down uniformly from \(20.0 \mathrm{~m} / \mathrm{s}\) to \(5.00 \mathrm{~m} / \mathrm{s}\) in \(4.00 \mathrm{~s}\). (a) What average force acted on the car during that time, and (b) how far did the car travel during that time?

A student decides to move a box of books into her dormitory room by pulling on a rope attached to the box. She pulls with a force of \(80.0 \mathrm{~N}\) at an angle of \(25.0^{\circ}\) above the horizontal. The box has a mass of \(25.0 \mathrm{~kg}\), and the coefficient of kinetic friction between box and floor is \(0.300 .\) (a) Find the acceleration of the box. (b) The student now starts moving the box up a \(10.0^{\circ}\) incline, keeping her \(80.0 \mathrm{~N}\) force directed at \(25.0^{\circ}\) above the line of the incline. If the coefficient of friction is unchanged, what is the new acceleration of the box?
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