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Chapter 7: Problem 5

State the definition of work done by a constant force.

Short Answer

Expert verified
The work done by a constant force is the dot product of the force acting on an object and the displacement of the object due to the force. It's measured using the formula \( W = Fd \cos(\theta) \), where W is the work, F is the force, d is the displacement, and \( \theta \) is the angle between the force and displacement vectors.

Step by step solution

01

Understanding Basic Concepts

Firstly, to understand the definition of work done by a constant force, one must have a basic understanding of the concepts of force, displacement, and the angle in between, within the perspective of physics.
02

Defining Force

Force is a vector quantity that describes an interaction between objects that changes the motion of an object. It has a magnitude, which signifies how strong the force is, and a direction, which shows where the force is acting towards.
03

Defining Displacement

Displacement, also a vector quantity, is an object's overall change in position. It's a straight-line distance in a specific direction, from the starting position to the final position.
04

Understanding Angle in Physics

The angle in physics is typically measured from the positive x-axis and is used to specify the direction of the vector quantities like force and displacement.
05

Defining Work Done by Constant Force

Now, the work done by a constant force is defined as the dot product of the force acting on an object and the displacement of the object due to that force. It can be measured using the formula \( W = Fd \cos(\theta) \) where W is the work done, F is the force, d is the displacement, and \( \theta \) is the angle between the force and displacement vectors.

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Key Concepts

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

Force
In the world of physics, force plays a crucial role. It is a vector quantity, meaning it has both magnitude and direction. Imagine a push or pull on an object鈥攖his is what we refer to as force. It can cause an object to start moving, stop moving, or change direction. For example:
  • Gravitational force pulls objects toward the Earth.
  • Frictional force resists motion between surfaces in contact.
Understanding force requires both numerical understanding (how strong it is) and spatial understanding (which way it acts). There are different types of forces in physics, such as contact forces and field forces, each behaving uniquely.
Displacement
Displacement is another vector quantity, essential in understanding motion. It differs from distance because it considers direction. Displacement measures the shortest straight-line distance from the starting point to the endpoint of an object's motion. Imagine walking in a zig-zag pattern鈥攜our distance is the entire path walked, but your displacement is a straight line from start to finishing point.
Think of it as:
  • Showing not just how far, but in which direction.
  • Always being a straight line, not the path traveled.
It鈥檚 a key concept when analyzing movement in physics, particularly when calculating work.
Dot Product
The dot product is a mathematical operation that combines two vectors. Unlike regular multiplication, the dot product considers the direction of the vectors and results in a scalar (a single number). It is calculated as follows:\[\text{Dot Product} = \mathbf{A} \cdot \mathbf{B} = AB \cos(\theta)\]Where:
  • \( A \) and \( B \) are the magnitudes of the vectors.
  • \( \theta \) is the angle between the vectors.
In the context of work done by a force, this operation helps determine how much of the force contributes to moving an object in the direction of displacement. It's a fundamental concept that connects all nodes of force, displacement, and angle in physics.
Angle in Physics
The angle in physics describes the orientation between two vectors, such as force and displacement. Measured from the positive x-axis, it dictates how much of one vector affects another. Consider an angle \( \theta \) between force and displacement:
  • If \( \theta = 0^\circ \), force and displacement align perfectly; all the force contributes to work.
  • If \( \theta = 90^\circ \), force is perpendicular to displacement; no work is done.
Understanding angles helps determine the effectiveness of a force in doing work on an object. This makes it key in the formula for work, where different angles affect how force produces movement.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the area of the region bounded by the graphs of \(f\) and \(g\) is 1, then the area of the region bounded by the graphs of \(h(x)=f(x)+C\) and \(k(x)=g(x)+C\) is also \(1 .\)

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