Question

A particle is subjected to three forces, ( F_1 ), ( F_2 ), and ( F_3 ). Forces ( F_1 ) and ( F_2 ) are defined with respect to the x and y axes as: (textmagnitude of F_1 ), ( textmagnitude of F_2 ).

a) ( F_3 )
b) ( F_1 + F_2 )
c) ( F_1 - F_2 )
d) ( F_2 - F_1 )

Answer 1

a)
`\( F_3 \)` is the resultant force when forces
`\( F_1 \)` and
`\( F_2 \)` are combined, calculated using vector addition.

Step-by-step explanation:

The resultant force
`\( F_3 \)` is determined by vector addition, where the magnitudes and directions of \( F_1 \) and
`\( F_2 \)` are considered. The formula for vector addition is
`\( F_3 = \sqrt{F_(1x)^2 + F_(2x)^2 + F_(1y)^2 + F_(2y)^2} \)`, where
`\( F_(1x) \) and \( F_(2x) \)` are the x-components of
`\( F_1 \)` and
`\( F_2 \)` respectively, and
`\( F_(1y) \)` and
`\( F_(2y) \)` are the y-components. This yields the magnitude of
`\( F_3 \)`.

To find the direction, the angle
`\( \theta \)` can be calculated using
`\( \theta = \tan^(-1)\left((F_(1y) + F_(2y))/(F_(1x) + F_(2x))\right) \).`

b)
`\( F_1 + F_2 \)` is the vector sum of
`\( F_1 \)` and
`\( F_2 \)`, calculated by adding their respective x and y components.

c)
`\( F_1 - F_2 \)` is the vector difference between
`\( F_1 \)` and
`\( F_2 \)`, obtained by subtracting their x and y components.

d)
`\( F_2 - F_1 \)` is the reverse vector difference, obtained by subtracting
`\( F_1 \)`'s components from
`\( F_2 \)`'s.

In summary, vector addition and subtraction involve combining or separating forces along the x and y axes, and the resulting vectors provide the magnitude and direction of the net force. These principles are fundamental in physics for analyzing and understanding the motion of particles under the influence of multiple forces.