NTBC-CT15: FISHINQ ROD-WEXHT OF TWO PIECES. An angler balances a fisking rod on her finger as shown. If she were to cut the rod along the dashed line, weeld the ucight of the picce on the left-hand side be greater Dhan, Los flusu, or equal to the weight of the piece on the right-hand side? Fivplain your reasoning.
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The fishing rod is balanced on her finger, which means the torques on both sides of the finger are equal. Show more…
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A thin rod of length $l$ lies on the $+x$ -axis with its left end at the origin. A string pulls on the rod with a force $\overrightarrow{\boldsymbol{F}}$ directed toward a point $P$ a distance $h$ above the rod. Where along the rod should you attach the string to get the greatest torque about the origin if point $P$ is (a) above the right end of the rod? (b) Above the left end of the rod? (c) Above the center of the rod?
Since, motion of the rod is purely translational, net toryue about the C.M. of the rod should be equal to zero. Thus $$ F_{1} \frac{l}{2}=F_{2}\left(\frac{l}{2}-a\right) \text { or, } \frac{F_{1}}{F_{2}}=1-\frac{a}{l / 2} $$ For the translational motion of rod. $$ F_{2}-F_{1}=m w_{c} \text { or } 1-\frac{F_{1}}{F_{2}}=\frac{m w_{c}}{F_{2}} $$ From (1) and (2) $$ \frac{a}{l / 2}=\frac{m w_{c}}{F_{2}} \text { or, } l=\frac{2 a F_{2}}{m w_{c}}=1 \mathrm{~m} $$
Physical Fundamentals Of Mdchanics
Dynamics of a Solid Body
Consider a metal rod of length $L$ fastened at both ends. If you cut the rod and weld on an additional segment of length $m,$ leaving the ends fixed, the rod will bow up into a circular arc of radius $R$ (unknown), as indicated in Figure $12 .$ Let $h$ be the maximum vertical displacement of the rod. (a) Show that $L=2 R \sin \theta$ and conclude that $$ h=\frac{L(1-\cos \theta)}{2 \sin \theta} $$ (b) Show that $L+m=2 R \theta$ and then prove $$ \frac{\sin \theta}{\theta}=\frac{L}{L+m} $$
APPLICATIONS OF THE DERIVATIVE
Newton's Method
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