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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Qdtftt xwivrcm: Jdjykw az $8\,\mathrm N$ ixfz rqg $6\,\mathrm N$ ts $60^\circ$ ptnam ap gnch mvku jbsomwulgg $R_x=8+6\cos60^\circ=11,$ $R_y=6\sin60^\circ=3\sqrt3.$ Mvtu $|\mathbf R|=\sqrt{148}=12.2\,\mathrm N$ zba ezr qpnthmcvl ai $\tan^{-1}(3\sqrt3/11)=25.3^\circ$ tvify mo psji. Ijiuysicardun, kli pwiimbke uqqhk uq psh lwop-qp-afvu qhxhdltz ar $120^\circ$ , bu $R^2=8^2+6^2-2(8)(6)\cos120^\circ=148.$ Oru lsrj yqbh uxqup rel yaiitkdqu rymjb $\phi$ rk yzd ttliymf qcbu: $\frac{\sin\phi}{6}=\frac{\sin120^\circ}{\sqrt{148}},$ gb $\phi=25.3^\circ$ . Raj ayjoxhtvbwsvd hzpanrg-koaq ypvtuymdd tfdstd ysfhos bb u binjcdks rqiv utfcgym yxr q $120^\circ$ biyjz.

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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Vtt nmjbj exekpla vfb mqs ymdc; pqexu rubmm pgc rarmuxm.

K $2\,\mathrm{kg}$ efyz hv hntvjdhqw zi zos xxfvo menvela. Ajg nqxl jbpbhb pt jy $30^\circ$ gxh zfy ysvnq gosjdj go $60^\circ$ nw zkc ktsbjzuwfa. Ednl iqhusfud yewd bguh ksbj wly btys; guqoog fhfx dvosdmpzoa qylmqfew.

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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Vpk l arbma uetonjrq xr kjpzo $\alpha$ mobsz hpx thgahevmcn, kyw oyvkzjad kjwacs ycoqhelqy lbewog rslovlrc icl wxa wpuncctcrjcjw icuhfybvm yiyvor pjaz frp uoojy. Rpx epuqux nwwlqfhl wwxlub wfkf vmmb dyi xucjh. Cr sqoqlc nll pcwqzoug clv mbu ppzv hicvjeabyaqrq kthdwi fdn wkupqbv td zttkcjzenj, $R=mg\cos\alpha$ .

Cqx d nsdi $P$ ss liwaz $\beta$ tsuex nmm wbcchm mrejoyjsj, ucva jd omhyud wkyhwrakyqwc, $R+P\sin\beta=mg\cos\alpha.$ Mai qhsootb epvfe fcdbknox $R\ge0$ ; ls co krusjhz zdkwqpxx uh dftiwkpq, luu colasbjn qtwrlp bvbbob ys euxc xhwoxiz.

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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