A metal bar is in the xyxy-plane with one end of the bar at the origin. A force F→=(7.00N)i+(−3.00N)j\overrightarrow{F}=\left(7.00N\right)i+(-3.00N)j is applied to the bar at the point x=3.00 mx=3.00\text{ m}, y=4.00 my=4.00\text{ m}. In terms of unit vectors ii and jj, what is the position vector rr for the point where the force is applied?

Ch 10: Dynamics of Rotational Motion

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

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Young & Freedman Calc 15th Edition

Ch 10: Dynamics of Rotational Motion

Problem 6a

Chapter 10, Problem 6a

A metal bar is in the $x y$ -plane with one end of the bar at the origin. A force $\vec{F} = (7.00 N) i + (- 3.00 N) j$ is applied to the bar at the point $x equals 3.00 m$ , $y equals 4.00 m$ . In terms of unit vectors $i$ and $j$ , what is the position vector $r$ for the point where the force is applied?

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Identify the coordinates of the point where the force is applied. The point is given as (x = 3.00 m, y = 4.00 m).

Understand that the position vector $ \mathbf{r} $ is a vector that points from the origin to the point where the force is applied. It is expressed in terms of unit vectors $ \mathbf{i} $ and $ \mathbf{j} $.

Write the position vector $ \mathbf{r} $ using the coordinates of the point. The position vector $ \mathbf{r} $ can be expressed as $ \mathbf{r} = x \mathbf{i} + y \mathbf{j} $.

Substitute the given values of x and y into the position vector equation. This gives $ \mathbf{r} = 3.00 \mathbf{i} + 4.00 \mathbf{j} $.

Conclude that the position vector $ \mathbf{r} $ for the point where the force is applied is $ 3.00 \mathbf{i} + 4.00 \mathbf{j} $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Position Vector

A position vector describes the location of a point in space relative to an origin. In the xy-plane, it is expressed using unit vectors i and j, representing the x and y components, respectively. For a point at coordinates (x, y), the position vector r is given by r = xi + yj, where x and y are the respective distances from the origin along the x and y axes.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to indicate direction in a coordinate system. In the context of the xy-plane, i and j are unit vectors pointing in the direction of the positive x-axis and y-axis, respectively. They are fundamental in expressing vectors in terms of their components, allowing for easy manipulation and calculation in vector algebra.

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system. In the xy-plane, any vector can be decomposed into its x-component and y-component, represented by the unit vectors i and j. This decomposition simplifies calculations involving vectors, such as addition, subtraction, and finding magnitudes, by treating each component separately.

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Related Practice

Textbook Question

Calculate the torque (magnitude and direction) about point O due to the force F in each of the cases sketched in Fig. E10.1. In each case, both the force F and the rod lie in the plane of the page, the rod has length 4.00 m, and the force has magnitude F = 10.0 N.

Textbook Question

A metal bar is in the $xy$ -plane with one end of the bar at the origin. A force $$\overrightarrow{F}$=$\left$(7.00N$\right$)i+(-3.00N)j$ is applied to the bar at the point $x=3.00$\text{ m}$$ , $y=4.00$\text{ m}$$ . What are the magnitude and direction of the torque with respect to the origin produced by $\vec{F}$ ?

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Textbook Question

One force acting on a machine part is F = (-5.00 N)i + (4.00 N)j. The vector from the origin to the point where the force is applied is r = (-0.450 m)i +(0.150 m)j. In a sketch, show r, F, and the origin.

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Textbook Question

A machinist is using a wrench to loosen a nut. The wrench is 25.0 cm long, and he exerts a 17.0-N force at the end of the handle at 37° with the handle (Fig. E10.7). What is the maximum torque he could exert with this force, and how should the force be oriented?

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Textbook Question

A machinist is using a wrench to loosen a nut. The wrench is 25.0 cm long, and he exerts a 17.0-N force at the end of the handle at 37° with the handle (Fig. E10.7). What torque does the machinist exert about the center of the nut?

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