Grade 11Mechanics

Two vectors have equal magnitudes of 12.7 units. They are oriented as shown in Fig. and their vector sum is . Find (a) the x and)' components of . (b) the magnitude of , and (c) the angle makes with the +x axis

Simran Bhatia

11 Years agoGrade 11

Aditi Chauhan

11 Years ago

Assumptions:
Let us assume that the vectors are given as:

We assume that the magnitude of vectors above are given as a , b and r respectively.
We also assume that the angle made by vector $\overrightarrow{a}$ from positive x axis, measured anticlockwise is given by $\theta _{a}$ whereas the angle made by vector $\overrightarrow{a}$ from positive x axis, measured counterclockwise is $\o$ .
Given:

First we write the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ in unit vector notations so that we can calculate the vector $\overrightarrow{r}$ using simple addition of their components
The figure below shows the vector diagram, representing vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ respectively.
It can be seen from the figure that the angle (say $\theta _{b}$ ) made by vector $\overrightarrow{b}$ from line AB, measured counterclockwise, is:

One should note that the above figure is not subjected to accurate scaling.
Therefore we can write the vector components of vector $\overrightarrow{b}$ as:

The negative sign in the vector component is due to the fact that the vector component $b_{x}\widehat{i}$ points in the direction opposite to that with the unit vector $\widehat{i}$ along the positive x axis.
Therefore vector $\overrightarrow{b}$ is:

If the angle made by vector $\overrightarrow{a}$ with respect to positive x axis, measured counterclockwise is $\theta _{a}$ , we can write the vector components of the vector as:

(a) The vector $\overrightarrow{r}$ can now be given as:

Substituting the value of components of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ from equation (1) and (2), we have:

Now for $\theta _{a} =$ 28.2°, $\theta _{b}$ = 46.8° , a = 12.7 units and b = 12.7 units, we can calculate the vector $\overrightarrow{r}$ using equation (3). Since the magnitude of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are equal, we can modify the equation (3) using a = b as:

One must note that we have just substituted the magnitude of vector $\overrightarrow{a}$ with that of vector $\overrightarrow{b}$ to obtain the equation above.
Now we substitute the given values of, $\theta _{a}$ = 28.2°, b = 12.7 units and calculated value of $\theta _{b}$ = 46.8° in equation (4) to obtain the vector $\overrightarrow{r}$ as:

From calculated value of vector $\overrightarrow{r}$ , we have

Therefore the horizontal component of vector $\overrightarrow{r}$ is 2.54 units.
The vertical component of vector $\overrightarrow{r}$ is 15.1 units.
(b) The magnitude of vector $\overrightarrow{r}$ is:

Therefore the magnitude of vector $\overrightarrow{r}$ is 15.3 units
(c) The angle made by vector $\overrightarrow{r}$ with positive x axis, measured counterclockwise can be given as:

Therefore the angle subtended by vector $\overrightarrow{r}$ is 80.4°with positive x axis, measured counterclockwise.

Two vectors have equal magnitudes of 12.7 units. They are oriented as shown in Fig. and their vector sum is . Find (a) the x and)' components of . (b) the magnitude of , and (c) the angle makes with the +x axis

Simran Bhatia

1 Answer

Aditi Chauhan

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If a=x² find the displacement at t =3sec if body velocity at t= 0 sec is 0

Two vectors have equal magnitudes of 12.7 units. They are oriented as shown in Fig. and their vector sum is . Find (a) the x and)' components of . (b) the magnitude of , and (c) the angle makes with the +x axis

Simran Bhatia

1 Answer

Aditi Chauhan

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