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Chapter 13: Problem 10

Explain how the work done by a force in moving an object is computed using dot products.

Short Answer

Expert verified
Answer: The work done by a force in moving an object is computed using the dot product of the force vector and the displacement vector. The dot product is calculated as \(\vec{F}\cdot\vec{d} = |\vec{F}||\vec{d}|\cos\theta\), where \(|\vec{F}|\) and \(|\vec{d}|\) are the magnitudes of the force and displacement vectors, and \(\theta\) is the angle between them. The result of the dot product represents the work done by the force in moving the object.

Step by step solution

01

Understanding Vectors and Force

A vector is a mathematical representation of a quantity possessing both magnitude and direction. In this context, we have two vectors: force (\(\vec{F}\)) and displacement (\(\vec{d}\)). Force is the push or pull exerted on an object, which results in a change in its motion or state of rest. Displacement represents the change in position of the object caused by the applied force.
02

Dot Product

The dot product (also known as the scalar product) is an operation performed between two vectors, resulting in a scalar value. Given two vectors \(\vec{A}\) and \(\vec{B}\), the dot product \(\vec{A} \cdot \vec{B}\) can be calculated by: \(\vec{A}\cdot\vec{B} = |\vec{A}||\vec{B}|\cos\theta\) where \(|\vec{A}|\) and \(|\vec{B}|\) are the magnitudes of the vectors, and \(\theta\) is the angle between them.
03

Work Done By a Force

Work done, represented as \(W\), is the energy transferred due to the applied force on an object. When a force moves an object, work is done when the force and displacement have some component in the same direction. The work done by a force (\(\vec{F}\)) in moving an object through a displacement (\(\vec{d}\)) is given by: \(W = \vec{F} \cdot \vec{d}\) This represents the dot product of the force and displacement vectors.
04

Calculating Work Done Using Dot Product

To calculate the work done using dot product, follow these steps: 1. Determine the force vector \(\vec{F}\) and displacement vector \(\vec{d}\). These vectors should be in the same unit of measurement (e.g., meters, feet, etc.). 2. Compute the magnitudes of force vector \(|\vec{F}|\) and displacement vector \(|\vec{d}|\). This can be done using the Pythagorean theorem for 2-dimensional vectors, or an extension of the same for 3-dimensional vectors. 3. Determine the angle \(\theta\) between force vector \(\vec{F}\) and displacement vector \(\vec{d}\). 4. Calculate the dot product \(\vec{F} \cdot \vec{d}\) using the formula: \(\vec{F}\cdot\vec{d} = |\vec{F}||\vec{d}|\cos\theta\) 5. The result of the dot product is the work done by the force in moving the object: \(W = \vec{F} \cdot \vec{d}\) The solution demonstrates the computation of work done by a force in moving an object using dot products, addressing the key concepts of vectors, dot product, and work.

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Key Concepts

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

Vectors
Vectors are fundamental components in mathematics and physics. They are essential for representing quantities that have both direction and magnitude. Unlike simple numbers that only have a size (known as scalars), vectors give us a complete picture of physical quantities. This is crucial when dealing with forces and movements in space.

Commonly, vectors in a problem are represented in arrow notation, where the arrow points in the direction of the vector and has a length proportional to the vector's magnitude.

To find the components of a vector, you break it down into perpendicular parts, usually along the x and y axes in a 2D plane. For example, a vector \(\vec{V}\) with components can be written as \((v_x, v_y)\), indicating its influence along both axes. This breakdown is essential when performing mathematical operations with vectors, such as addition, subtraction, and, importantly for this topic, the dot product.
Force
Force is a vector quantity. It measures the push or pull exerted on an object and possesses both magnitude and direction. Forces cause objects to move, stop, or change their velocity. In physics, the symbol \(\vec{F}\) is commonly used to denote force vectors.

Forces can be as simple as the force of gravity pulling an object towards the Earth, or more complex, involving friction, tension, and other types. Forces acting on an object can be calculated individually or in combination for a result known as the net force.

Understanding and calculating force is key to determining the work done on an object. Since force has both direction and magnitude, it can be represented as a vector, \(\vec{F}\), in vector operations like the dot product. This is critical when calculating the work done by the force, as the direction of the force relative to the movement of the object is directly related to how much work is done.
Work Done
Work in physics is a measure of energy transfer when a force moves an object over a distance. This concept is directly tied to energy, as doing work on an object increases its energy or converts potential energy into kinetic energy.

The formula for work done, \(W = \vec{F} \cdot \vec{d}\), uses the dot product of force \(\vec{F}\) and displacement \(\vec{d}\). The outcome is a scalar, representing the total energy transferred as a result of the movement.

A critical aspect of work done is that only the component of the force that acts in the direction of the displacement contributes to the work. If the force and displacement are in the same direction, maximum work is done; if they are perpendicular, no work is done. This is why the dot product, which considers the cosine of the angle between the force and displacement, is used: it effectively measures the alignment of the vectors.

Using the formula, work can be calculated whether the vectors provide direct numbers or require breakdown into components, ensuring that work calculations are accurate in diverse physics problems.

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Most popular questions from this chapter

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