Chapter 1: Q6E (page 329)

A metal bar is in the xy plane with one end of the bar at the origin. A force $\vec{F} = (7.00 N) \hat{i} + (- 3.00 N) \hat{j}$ is applied to the bar at the point $x = 3.00 m, y = 4.00 m$ . (a) In terms of unit vectors $\hat{i}$ and $\hat{j}$ , what is the position vector $\vec{r}$ for the point where the force is applied? (b) What are the magnitude and direction of the torque with respect to the origin produced by $\vec{F}$ ?

Short Answer

Expert verified

(a) the position vector in terms of unit vectors $\hat{i}, \hat{j}$ is $\vec{r} = (3.00) \hat{i} + (4.00) \hat{j}$ .

(b) The torque with the magnitude and direction is,

$τ = 37.0 N \cdot m$ in -z direction.

Step by step solution

To mention the given data

We have the given data:

The force given is, $\vec{F} = (7.00 N) \hat{i} + (- 3.00 N) \hat{j}$ .

The points where the force is applied: $x = 3.00, y = 4.00$ .

To recall the concepts

The position vector in terms of unit vectors $\hat{i}, \hat{j}$ is given by,

$\vec{r} = x \hat{i} + y \hat{j}$ .

The torque is given by,

$\vec{τ} = \vec{r} \times \vec{F}$ .

(a)To find the position vector

The position vector in terms of unit vectors $\hat{i}, \hat{j}$ is given by,

$\vec{r} = x \hat{i} + y \hat{j} \Rightarrow \vec{r} = (3.00) \hat{i} + (4.00) \hat{j}$

Hence,the position vector in terms of unit vectors $\hat{i}, \hat{j}$ is $\vec{r} = (3.00) \hat{i} + (4.00) \hat{j}$ .

(b)To find the magnitude and direction of the torque

The torque is given by,

$\vec{τ} = \vec{r} \times \vec{F}$ .

Substituting the values, we get,

$\vec{τ} = ((3.00) \hat{i} + (4.00) \hat{j}) \times ((7.00) \hat{i} + (- 3.00) \hat{j})$

We will use the equation 1.25 from the Chapter 1, we get,

$\vec{τ} = (- 9.00 N \cdot m) \hat{k} + (- 28.00 N \cdot m) (- \hat{k}) = (- 37.0 N \cdot m) \hat{k}$

Hence, the torque with the magnitude and direction is,

$τ = 37.0 N \cdot m$ in -z direction.

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Short Answer

Step by step solution

To mention the given data

To recall the concepts

(a)To find the position vector

(b)To find the magnitude and direction of the torque

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