Resolving Forces & Inclines

01Resolving Forces & Inclines

Resolving Forces

Replace a force by signed components along perpendicular axes.

Measure the angle from the chosen axis

F_x=F\cos\theta,\qquad F_y=F\sin\theta

Here $\theta$ is measured anticlockwise from the positive $x$ -axis. Components can be negative. If the given angle is measured from the vertical, the adjacent component is vertical instead. Resolving replaces the force: do not include the original force and its components in the same sum.

Components replace the original force in an equation.

The force is the hypotenuse of a component triangle. Use cosine for the component adjacent to the marked angle, and sine for the opposite component. Give each component its directional sign.

02Resolving Forces & Inclines

Resultants & Force Triangles

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The resultant joins the start to the finish

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03Resolving Forces & Inclines

Equilibrium in 2D

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Balance forces in two independent directions

\sum F_x=0,\qquad \sum F_y=0

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04Resolving Forces & Inclines

Forces on Inclines

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Use axes along and normal to the plane

W_{\parallel}=mg\sin\alpha,\qquad W_{\perp}=mg\cos\alpha

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05Resolving Forces & Inclines

Motion with Resolved Forces

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Find the reaction separately from the acceleration

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P\cos\beta-mg\sin\alpha=ma

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