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Question Number 226442 by mr W last updated on 28/Nov/25
Commented by mahdipoor last updated on 29/Nov/25
for mass M ΣF=ma (for CM) { ((A=(x^(..) −α.l.cosθ−ω^2 .l.sinθ)M)),((B=(−α.l.sinθ+ω^2 .l.cosθ)M)) :} ΣT=Iα (for CM) { ((Alsinθ+Blcosθ=0 )) :} for mass 3M { ((−A=(x^(..) )3M)),((N_(rod) −B=0)) :}
$$\mathrm{for}\:\mathrm{mass}\:\mathrm{M} \\ $$$$\Sigma\mathrm{F}=\mathrm{ma}\:\:\left(\mathrm{for}\:\mathrm{CM}\right) \\ $$$$\begin{cases}{\mathrm{A}=\left(\overset{..} {\mathrm{x}}−\alpha.\mathrm{l}.\mathrm{cos}\theta−\omega^{\mathrm{2}} .\mathrm{l}.\mathrm{sin}\theta\right)\mathrm{M}}\\{\mathrm{B}=\left(−\alpha.\mathrm{l}.\mathrm{sin}\theta+\omega^{\mathrm{2}} .\mathrm{l}.\mathrm{cos}\theta\right)\mathrm{M}}\end{cases} \\ $$$$\Sigma\mathrm{T}=\mathrm{I}\alpha\:\:\:\left(\mathrm{for}\:\mathrm{CM}\right) \\ $$$$\begin{cases}{\mathrm{Alsin}\theta+\mathrm{Blcos}\theta=\mathrm{0}\:}\end{cases} \\ $$$$\mathrm{for}\:\mathrm{mass}\:\mathrm{3M} \\ $$$$\begin{cases}{−\mathrm{A}=\left(\overset{..} {\mathrm{x}}\right)\mathrm{3M}}\\{\mathrm{N}_{\mathrm{rod}} −\mathrm{B}=\mathrm{0}}\end{cases} \\ $$
Answered by mr W last updated on 29/Nov/25
Commented by mr W last updated on 29/Nov/25
μ=(m/M)=((3M)/M)=3 let ω=−(dθ/dt) u=lω v_x =−V+u cos θ (←) v_y =u sin θ (↓) conservation of horizontal momentum mV=M(−V+u cos θ) ⇒V=((lω cos θ)/(μ+1)) conservation of energy: ((mV^2 )/2)+((M[(−V+u cos θ)^2 +(u sin θ)^2 ])/2)=((Mgl)/2)(cos θ−cos θ_0 ) (1+μ)V^2 +u^2 −2Vu cos θ=gl(cos θ−cos θ_0 ) lω^2 (μ+sin^2 θ)=g(μ+1)(cos θ−cos θ_0 ) ⇒ω=−(dθ/dt)=(√((g(μ+1)(cos θ−cos θ_0 ))/(l(μ+sin^2 θ)))) dt=−(√((l(μ+sin^2 θ))/(g(μ+1)(cos θ−cos θ_0 )))) dθ ⇒t=(√(l/(g(μ+1))))∫_θ ^θ_0 (√((μ+sin^2 θ)/(cos θ−cos θ_0 ))) dθ time period of a full oscillation: T=4(√(l/(g(μ+1))))∫_0 ^θ_0 (√((μ+sin^2 θ)/(cos θ−cos θ_0 ))) dθ example: μ=(m/M)=3, θ_0 =(π/3) T=2(√(l/g))∫_0 ^(π/3) (√((3+sin^2 θ)/(cos θ−(1/2)))) dθ ≈8.810467(√(l/g)) V=((lω cos θ)/(μ+1)) A=(dV/dt)=−ω(dV/dθ)=−((lω)/(μ+1))(cos θ (dω/dθ)−ω sin θ) T sin θ=mA=−((mlω)/(μ+1))(cos θ (dω/dθ)−ω sin θ) N=mg+T cos θ=mg−((mlω cos θ)/((μ+1)sin θ))(cos θ (dω/dθ)−ω sin θ) (N/(mg))=1−((l cos θ)/(g(μ+1)sin θ))(cos θ ((ωdω)/dθ)−ω^2 sin θ)
$$\mu=\frac{{m}}{{M}}=\frac{\mathrm{3}{M}}{{M}}=\mathrm{3} \\ $$$${let}\:\omega=−\frac{{d}\theta}{{dt}} \\ $$$${u}={l}\omega \\ $$$${v}_{{x}} =−{V}+{u}\:\mathrm{cos}\:\theta\:\:\:\:\left(\leftarrow\right) \\ $$$${v}_{{y}} ={u}\:\mathrm{sin}\:\theta\:\:\:\:\left(\downarrow\right) \\ $$$${conservation}\:{of}\:{horizontal}\:{momentum}\: \\ $$$${mV}={M}\left(−{V}+{u}\:\mathrm{cos}\:\theta\right) \\ $$$$\Rightarrow{V}=\frac{{l}\omega\:\mathrm{cos}\:\theta}{\mu+\mathrm{1}} \\ $$$${conservation}\:{of}\:{energy}: \\ $$$$\frac{{mV}^{\mathrm{2}} }{\mathrm{2}}+\frac{{M}\left[\left(−{V}+{u}\:\mathrm{cos}\:\theta\right)^{\mathrm{2}} +\left({u}\:\mathrm{sin}\:\theta\right)^{\mathrm{2}} \right]}{\mathrm{2}}=\frac{{Mgl}}{\mathrm{2}}\left(\mathrm{cos}\:\theta−\mathrm{cos}\:\theta_{\mathrm{0}} \right) \\ $$$$\left(\mathrm{1}+\mu\right){V}^{\mathrm{2}} +{u}^{\mathrm{2}} −\mathrm{2}{Vu}\:\mathrm{cos}\:\theta={gl}\left(\mathrm{cos}\:\theta−\mathrm{cos}\:\theta_{\mathrm{0}} \right) \\ $$$${l}\omega^{\mathrm{2}} \left(\mu+\mathrm{sin}^{\mathrm{2}} \:\theta\right)={g}\left(\mu+\mathrm{1}\right)\left(\mathrm{cos}\:\theta−\mathrm{cos}\:\theta_{\mathrm{0}} \right) \\ $$$$\Rightarrow\omega=−\frac{{d}\theta}{{dt}}=\sqrt{\frac{{g}\left(\mu+\mathrm{1}\right)\left(\mathrm{cos}\:\theta−\mathrm{cos}\:\theta_{\mathrm{0}} \right)}{{l}\left(\mu+\mathrm{sin}^{\mathrm{2}} \:\theta\right)}} \\ $$$${dt}=−\sqrt{\frac{{l}\left(\mu+\mathrm{sin}^{\mathrm{2}} \:\theta\right)}{{g}\left(\mu+\mathrm{1}\right)\left(\mathrm{cos}\:\theta−\mathrm{cos}\:\theta_{\mathrm{0}} \right)}}\:{d}\theta \\ $$$$\Rightarrow{t}=\sqrt{\frac{{l}}{{g}\left(\mu+\mathrm{1}\right)}}\int_{\theta} ^{\theta_{\mathrm{0}} } \sqrt{\frac{\mu+\mathrm{sin}^{\mathrm{2}} \:\theta}{\mathrm{cos}\:\theta−\mathrm{cos}\:\theta_{\mathrm{0}} }}\:{d}\theta \\ $$$${time}\:{period}\:{of}\:{a}\:{full}\:{oscillation}: \\ $$$${T}=\mathrm{4}\sqrt{\frac{{l}}{{g}\left(\mu+\mathrm{1}\right)}}\int_{\mathrm{0}} ^{\theta_{\mathrm{0}} } \sqrt{\frac{\mu+\mathrm{sin}^{\mathrm{2}} \:\theta}{\mathrm{cos}\:\theta−\mathrm{cos}\:\theta_{\mathrm{0}} }}\:{d}\theta \\ $$$$ \\ $$$${example}: \\ $$$$\mu=\frac{{m}}{{M}}=\mathrm{3},\:\theta_{\mathrm{0}} =\frac{\pi}{\mathrm{3}} \\ $$$${T}=\mathrm{2}\sqrt{\frac{{l}}{{g}}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \sqrt{\frac{\mathrm{3}+\mathrm{sin}^{\mathrm{2}} \:\theta}{\mathrm{cos}\:\theta−\frac{\mathrm{1}}{\mathrm{2}}}}\:{d}\theta \\ $$$$\:\:\:\approx\mathrm{8}.\mathrm{810467}\sqrt{\frac{{l}}{{g}}} \\ $$$$ \\ $$$${V}=\frac{{l}\omega\:\mathrm{cos}\:\theta}{\mu+\mathrm{1}} \\ $$$${A}=\frac{{dV}}{{dt}}=−\omega\frac{{dV}}{{d}\theta}=−\frac{{l}\omega}{\mu+\mathrm{1}}\left(\mathrm{cos}\:\theta\:\frac{{d}\omega}{{d}\theta}−\omega\:\mathrm{sin}\:\theta\right) \\ $$$${T}\:\mathrm{sin}\:\theta={mA}=−\frac{{ml}\omega}{\mu+\mathrm{1}}\left(\mathrm{cos}\:\theta\:\frac{{d}\omega}{{d}\theta}−\omega\:\mathrm{sin}\:\theta\right) \\ $$$${N}={mg}+{T}\:\mathrm{cos}\:\theta={mg}−\frac{{ml}\omega\:\mathrm{cos}\:\theta}{\left(\mu+\mathrm{1}\right)\mathrm{sin}\:\theta}\left(\mathrm{cos}\:\theta\:\frac{{d}\omega}{{d}\theta}−\omega\:\mathrm{sin}\:\theta\right) \\ $$$$\frac{{N}}{{mg}}=\mathrm{1}−\frac{{l}\:\mathrm{cos}\:\theta}{{g}\left(\mu+\mathrm{1}\right)\mathrm{sin}\:\theta}\left(\mathrm{cos}\:\theta\:\frac{\omega{d}\omega}{{d}\theta}−\omega^{\mathrm{2}} \:\mathrm{sin}\:\theta\right) \\ $$

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