11판/12. 평형과 탄성

12-4 할리데이 11판 솔루션 일반물리학

짱세디럭스 2024. 6. 25. 17:45

(모든 블록의 y축 두께는 문제의 결론에 영향을 주지 않으므로 무시합니다.) (a)\ab{a} put {T:TopL:Middle LeftR:Middle RightB:Bottoml:length \put \begin{cases} T : \text{Top}\\ L : \text{Middle Left}\\ R : \text{Middle Right}\\ B : \text{Bottom}\\ l : \text{length} \end{cases} {ΣFT=0=NTmgΣFL=0=NLNTmgΣFR=0=NRmgΣFB=0=NBNLNRmg \begin{cases} \Sigma F_T&=0=N_T-mg\\ \Sigma F_{L}&=0=N_{L}-N_T-mg\\ \Sigma F_{R}&=0=N_{R}-mg\\ \Sigma F_B&=0=N_B-N_L-N_R-mg\\ \end{cases} {NT=mgNL=2mgNR=mgNB=4mg \begin{cases} N_T&=mg\\ N_{L}&=2mg\\ N_{R}&=mg\\ N_B&=4mg\\ \end{cases} [Stable Case]\title{Stable Case} l2com(Block,x)l2BlockStable -{l\over 2}\le \com(\text{Block},x)\le{l\over 2}\\ \Updownarrow\\ \text{Block}\rarr\text{Stable} [About T Block]\title{About T Block} com(T,x)=0, \com(T,x)=0, TStable,l20l2 T\rarr\text{Stable},\\ \Updownarrow\\ -{l\over 2}\le 0\le{l\over 2}\\ [About L+T Block]\title{About L+T Block} com(L+T,x)=l2+a1, \com(L+T,x)={-{l\over2}+a_1}, L+TStable,l2l+a1l2 L+T\rarr\text{Stable},\\ \Updownarrow\\ -{l\over 2}\le -l+a_1\le{l\over 2}\\ l2a132l(1)\therefore {l\over 2}\le a_1\le {3\over2}l\taag1 [About R Block]\title{About R Block} com(R,x)=a1, \com(R,x)={a_1}, RStable,l2a1l2 R\rarr\text{Stable},\\ \Updownarrow\\ -{l\over 2}\le a_1\le{l\over 2}\\ l2a1l2(2)\therefore -{l\over 2}\le a_1\le{l\over 2}\taag2 (1),(2)a1=l2(1),(2)\rarr a_1={l\over 2} [About All Block]\title{About All Block} com(All,x)=ΣxmΣm=02m+lm+l2m4m=38l \begin{aligned} \com(\text{All},x) &={\Sigma xm\over\Sigma m}\\ &={0\cdot 2m+l\cdot m+{l\over2}\cdot m\over4 m}\\ &={3\over8}l\\ \end{aligned} com(All,x)la238lla2a258l \begin{aligned} \com(\text{All},x)&\le l-a_2\\ {3\over8}l&\le l-a_2\\ a_2&\le{5\over8}l \end{aligned} ha=a1+a2=l2+58l=98l \begin{aligned} h_a &=a_1+a_2\\ &={l\over 2}+{5\over8}l\\ &={9\over8}l\\ \end{aligned} (b)\ab{b} put {T:TopL:Middle LeftR:Middle RightB:Bottom \put \begin{cases} T : \text{Top}\\ L : \text{Middle Left}\\ R : \text{Middle Right}\\ B : \text{Bottom}\\ \end{cases} {ΣFT=0=NTL+NTRmgΣFL=0=NLNTLmgΣFR=0=NRNTRmgΣFB=0=NBNLNRmgΣτL=0=l2mg+b1NLlNTLΣτR=0=l2mgb1NR+lNTR \begin{cases} \Sigma F_T&=0=N_{TL}+N_{TR}-mg\\ \Sigma F_{L}&=0=N_{L}-N_{TL}-mg\\ \Sigma F_{R}&=0=N_{R}-N_{TR}-mg\\ \Sigma F_B&=0=N_B-N_L-N_R-mg\\ \Sigma \tau_L&=0=-{l\over2}\cdot mg+b_1 \cdot N_L-l\cdot N_{TL}\\ \Sigma \tau_R&=0={l\over2}\cdot mg-b_1 \cdot N_R+l\cdot N_{TR}\\ \end{cases} {NTL=12mgNTR=12mgNL=32mgNR=32mgNB=4mgb1=23l \begin{cases} N_{TL}&={1\over2}mg\\ N_{TR}&={1\over2}mg\\ N_{L}&={3\over2}mg\\ N_R&={3\over2}mg\\ N_B&=4mg\\ b_1&={2\over3}l \end{cases} [About All Block]\title{About All Block} com(All,x)=l2lb2b2l2 \begin{aligned} \com(\text{All},x)={l\over2}&\le l-b_2\\ b_2&\le{l\over2} \end{aligned} hb=b1+b2=23l+l2=76l \begin{aligned} h_b&=b_1+b_2\\ &={2\over3}l+{l\over2}\\ &={7\over6}l\\ \end{aligned}