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Chapter 12: Problem 6

Explain how to find the torque produced by a force using cross products.

Short Answer

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Question: Explain how to find the torque produced by a force using cross products. Answer: To find the torque produced by a force using cross products, follow these steps: 1. Understand the concept of torque, which is the rotational equivalent of force and is measured in newton-meters (N路m). 2. Understand the concept of the cross product, a binary operation on vectors in a three-dimensional space that results in a vector perpendicular to the input vectors. 3. Identify the position vector (r) and the force vector (F). The position vector extends from the axis of rotation to the point where the force is applied, while the force vector represents the magnitude and direction of the applied force. 4. Calculate the cross product of the position vector (r) and the force vector (F): 蟿 = |r 脳 F|. 5. Find the magnitude of the cross product, which represents the torque produced by the force.

Step by step solution

01

Understand the concept of torque

Torque is the rotational equivalent of force. It measures the force's ability to cause an object to rotate around an axis or pivot point. It is represented by the symbol \(\tau\) and is measured in newton-meters (N路m). Torque depends not only on the magnitude of the applied force, but also on the distance between the axis of rotation (or pivot point) and the point where the force is applied on the object. The further the applied force is from the pivot point, the greater the torque.
02

Understand the concept of cross product

The cross product, also known as the vector product, is a binary operation on vectors in a three-dimensional space. Given two vectors \(\vec{A}\) and \(\vec{B}\), the cross product \(\vec{A} \times \vec{B}\) is a vector that is perpendicular to both \(\vec{A}\) and \(\vec{B}\). The magnitude of the cross product is equal to the product of the magnitudes of the two input vectors and the sine of the angle between them (\(|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}| \sin\theta\)).
03

Identify the position and force vectors

To find the torque produced by a force using cross products, we first need to identify the position vector (\(\vec{r}\)) and the force vector (\(\vec{F}\)). The position vector \(\vec{r}\) extends from the axis of rotation (or pivot point) to the point where the force is applied. The force vector \(\vec{F}\) represents the magnitude and direction of the applied force.
04

Calculate the cross product

The torque produced by a force can be found using the cross product of the position vector \(\vec{r}\) and the force vector \(\vec{F}\). Mathematically, we can represent this as: \(\tau = |\vec{r} \times \vec{F}|\). Specifically, if the position vector \(\vec{r} = \left\langle x_r,y_r,z_r \right\rangle\) and the force vector \(\vec{F} = \left\langle x_F,y_F,z_F \right\rangle\), then the cross product can be calculated as follows: $$ \vec{r} \times \vec{F} = \left\langle y_r z_F - z_r y_F, z_r x_F - x_r z_F, x_r y_F - y_r x_F \right\rangle. $$
05

Find the magnitude of the cross product

The cross product itself is a vector, but we are interested in finding the torque, which is the magnitude of the cross product. We can find the magnitude using the formula \(|\vec{r} \times \vec{F}| = \sqrt{(x_{r \times F})^2 + (y_{r \times F})^2 + (z_{r \times F})^2}\), where \(x_{r \times F}, y_{r \times F}, z_{r \times F}\) are the components of the cross product vector. By following these steps, we can find the torque produced by a force using cross products.

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