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Chapter 11: Problem 8

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

Short Answer

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Question: Explain how to compute the work done by a force in moving an object using dot products and provide the steps to calculate the work done. Answer: To compute the work done by a force in moving an object using dot products, we first need to understand the dot product operation. The dot product of two vectors A and B is defined as A 鈰� B = |A| |B| cos 胃, where 胃 is the angle between them. For calculating work done by a force (F) in moving an object in the direction of displacement (d), we use the formula W = F 鈰� d. To calculate the work done, follow these steps: 1. Determine the force vector (F) and displacement vector (d). 2. Calculate the magnitude of each vector (|F| and |d|). 3. Determine the angle 胃 between the force vector and the displacement vector. 4. Compute the cosine of the angle (cos 胃). 5. Use the dot product formula to calculate the work done: W = F 鈰� d = |F| |d| cos 胃.

Step by step solution

01

Reviewing the concept of dot products

The dot product, also called the scalar product, is an operation that takes two vectors and produces a scalar value. In general, the dot product of two vectors A and B is defined as the product of their magnitudes and the cosine of the angle between them. Mathematically, this is written as A 鈰� B = |A| |B| cos 胃.
02

Understanding the formula for work done

The work done (W) by a force (F) in moving an object in the direction of displacement (d) is given by the following formula: W = F 鈰� d. Here, the force and displacement are vector quantities, and the work done is a scalar quantity.
03

Explaining the dot product in the context of work done

The dot product can be used to compute the work done by a force when it moves an object in the direction of displacement. Since F and d are vectors, the dot product of the two vectors allows us to determine the scalar quantity of work done. Therefore, the work done can be calculated using the dot product formula: W = F 鈰� d = |F| |d| cos 胃, where 胃 is the angle between the force vector and the displacement vector.
04

Calculating the work done using dot products

To compute the work done using dot products, follow these steps: 1. Determine the force vector (F) and displacement vector (d). 2. Calculate the magnitude of each vector (|F| and |d|). 3. Determine the angle 胃 between the force vector and the displacement vector. 4. Compute the cosine of the angle (cos 胃). 5. Use the dot product formula to calculate the work done: W = F 鈰� d = |F| |d| cos 胃.

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