Question

Two forces A and B are 6N at 36° to the positive x-axis and 7N along the negative x-axisx-axis respectively. Calculate :
1) magnitude of A+B is ?
2) the direction of A+ B.
3) the magnitude of A - B

Answer 1

Answer on Question #68525Physics / Other

Two forces $\mathbf{A}$ and $\mathbf{B}$ are 6N at $36{}^{\circ}$ to the positive x-axis and 7N along the negative x-axis respectively. Calculate:

1) magnitude of $\mathbf{A} + \mathbf{B}$ is?

2) the direction of $\mathbf{A} + \mathbf{B}$ .

3) the magnitude of $\mathbf{A} - \mathbf{B}$ .

Solution:

1) The magnitude of $\mathbf{A} + \mathbf{B}$ is

$$| \mathbf {A} + \mathbf {B} | = \sqrt {(6 \cos 3 6 {}^ {\circ} - 7) ^ {2} + (6 \sin 3 6 {}^ {\circ}) ^ {2}} = 4. 1 \mathrm {N}$$

2) The direction of $\mathbf{A} + \mathbf{B}$ is

$$\left(\mathbf {A} + \mathbf {B}\right) _ {x} = 6 \cos 3 6 {}^ {\circ} - 7$$ $$\left(\mathbf {A} + \mathbf {B}\right) _ {y} = 6 \sin 3 6 {}^ {\circ}$$ $$\tan \varphi = \frac {(\mathbf {A} + \mathbf {B}) _ {y}}{(\mathbf {A} + \mathbf {B}) _ {x}} = \frac {6 \sin 3 6 {}^ {\circ}}{6 \cos 3 6 {}^ {\circ} - 7} = - 1. 6 4, \quad \varphi = \arctan (- 1. 6 4) = - 5 8. 6 {}^ {\circ}$$

Vector $\mathbf{A} + \mathbf{B}$ is directed at angle $\varphi = 58.6{}^{\circ}$ to the negative x-axis.

3) The magnitude of $\mathbf{A} - \mathbf{B}$ is

$$| \mathbf {A} - \mathbf {B} | = \sqrt {(6 \cos 3 6 {}^ {\circ} + 7) ^ {2} + (6 \sin 3 6 {}^ {\circ}) ^ {2}} = 1 2. 4 \mathrm {N}$$

Answers:

1) $|\mathbf{A} + \mathbf{B}| = 4.1\mathrm{N}$

2) $58.6{}^{\circ}$ to the negative x-axis

3) $|\mathbf{A} - \mathbf{B}| = 12.4\mathrm{N}$

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