Chapter 6.3, Problem 111E

(a)

To determine

To Find:The angle between the two forces

Answer to Problem 111E

If the magnitude of the resultant is equal to the sum of the magnitudes of the forces, then the angle between the forces is $0^{\circ }$

Explanation of Solution

Given information:

Two forces of equal magnitude acting on a point

Concept used: let $u=(u_1,u_2)$

and $v=(v_1,v_2)$

be vectors. Then the sum of u and v is the vector

  $u+v=(u_1+v_1,u_2+v_2)$

Calculation:

  $F_1=\left(F_1\cos \theta i+F_1\sin \theta j\right)$

  $F_2=\left(F_2\cos \theta i+F_2\sin \theta j\right)$

  $F_1+F_2=F_1\cos \theta i+F_1\sin \theta j+F_2\cos \theta i+F_2\sin \theta j$

  $\Vert F_1+F_2\Vert =\Vert \left(F_1+F_2\right)\cos \theta i+\left(F_1+F_2\right)\sin \theta j\Vert$

If the magnitude of the resultant is equal to the sum of the magnitudes of the forces, then the angle between the forces is 0°

  $\Vert F_1+F_2\Vert =\Vert \left(F_1+F_2\right)\cos 0^{\circ }i+\left(F_1+F_2\right)\sin 0^{\circ }j\Vert$

  $\Vert F_1+F_2\Vert =\Vert \left(F_1+F_2\right)i\Vert$

(b)

To determine

To Find:If the resultant of the forces is $0$ , the what will be the angle between the forces

Answer to Problem 111E

Resultant of the forces is equal to zero when the angle between the forces is $\pi$

Explanation of Solution

Given information:

The resultant of the forces is $0$

Concept used: let $u=(u_1,u_2)$

and $v=(v_1,v_2)$

be vectors. Then the sum of u and v is the vector

  $u+v=(u_1+v_1,u_2+v_2)$

Calculation:

  $F_1=\left(F_1\cos \theta i+F_1\sin \theta j\right)$

  $F_2=\left(F_2\cos \theta i+F_2\sin \theta j\right)$

  $F_1+F_2=F_1\cos \theta i+F_1\sin \theta j+F_2\cos \theta i+F_2\sin \theta j$

If the resultant of the forces is $0$

  $0=\left(F_1+F_2\right)\cos \theta i+\left(F_1+F_2\right)\sin \theta j$

Resultant of the forces is equal to zero when the angle between the forces is $\pi$

(c)

To determine

The magnitude of the resultant be greater than the sum of the magnitudes of the two forces

Answer to Problem 111E

the magnitude of the resultant cannot be greater than the sum of the magnitudes of the two forces

Explanation of Solution

Given information:

Two forces of equal magnitude acting on a point

Concept used: let $u=(u_1,u_2)$

And

  $v=(v_1,v_2)$ be vectors. Then the sum of u and v is the vector

  $u+v=(u_1+v_1,u_2+v_2)$

Calculation:

  $F_1=\left(F_1\cos \theta i+F_1\sin \theta j\right)$

  $F_2=\left(F_2\cos \theta i+F_2\sin \theta j\right)$

  $F_1+F_2=F_1\cos \theta i+F_1\sin \theta j+F_2\cos \theta i+F_2\sin \theta j$

  $F_1+F_2$

  $=\left(F_1+F_2\right)\cos \theta i+\left(F_1+F_2\right)\sin \theta j$

The magnitude of the resultant is equal to the sum of the magnitudes of the forces, when the angle between the forces is $0^{\circ }$

Therefore, the magnitude of the resultant cannot be greater than the sum of the magnitudes of the two forces