6.1 Spring Problems I
- 1
\(y=\dst{3 \cos4\sqrt{6}t-{1\over2\sqrt{6}} \sin 4\sqrt{6} t}\) ft
- 2
\(y=\dst{ -{1\over4}\cos8\sqrt{5}t-{1\over 4\sqrt{5}}\sin8\sqrt{5}t}\) ft
- 3
\(y=1.5\cos14\sqrt{10}t\) cm
- 4
\(y=\dst{{1\over4}\cos8t-{1\over16}\sin8t}\) ft; \(R=\dst{\sqrt{17}\over16}\) ft; \(\omega_0=8\) rad/s; \(T=\pi/4\) s;
\(\phi\approx-.245\mbox{ rad}\approx -14.04^\circ\);
- 5
\(y=\dst{10\cos14t+{25\over14}\sin14t}\) cm; \(R=\dst{5\over14}\sqrt{809}\) cm; \(\omega_0=14\) rad/s; \(T=\pi/7\) s;
\(\phi\approx.177\) rad \(\approx 10.12^\circ\)
- 6
\(y=\dst{-{1\over4}\cos\sqrt{70}\; t+{2 \over\sqrt{70}}\sin\sqrt{70}\; t}\) m; \(R= \dst{{1\over4}\sqrt{67\over 35}}\) m \(\omega_0=\sqrt{70}\) rad/s;
\(T=2\pi/\sqrt{70}\) s; \(\phi\approx 2.38\mbox{ rad}\approx 136.28^\circ\)
- 7
\(y=\dst{{2\over 3}\cos16t-{1\over4}\sin16t\mbox{ ft}}\)
- 8
\(y=\dst{{1\over2}\cos8t-{3\over8}\sin8t}\) ft
- 9
\(.72\mbox{ m}\)
- 10
\(y=\dst{{1\over 3}\sin t +{1\over2}\cos2t+{5\over6}\sin2t}\) ft
- 11
\(y=\dst{{16\over5}\left(4\sin {t\over4}-\sin t\right)}\)
- 12
\(y=\dst{-{1\over16} \sin8t+{1\over 3}\cos4\sqrt2 t -{1\over8\sqrt2 }\sin4\sqrt2 t}\)
- 13
\(y=\dst{-t\cos8t-{1\over6}\cos8t+{1\over8} \sin8t}\) ft
- 14
\(T=4\sqrt{2}\) s
- 15
\(\omega=8\) rad/s \(y=-\dst{{t\over 16}(-\cos8t+2\sin 8t)+{1\over128}\sin 8t}\) ft
- 16
\(\dst{\omega=4\sqrt6\mbox{ rad/s}; \quad y=-{t\over \sqrt{6}}\left[{8\over 3}\cos4\sqrt{6}t+4\sin 4\sqrt{6}t\right]+{1\over9}\sin 4\sqrt{6} t}\) ft
- 17
\(y=\dst{{t\over2}\cos2t-{t\over4}\sin2t+3\cos2t+2\sin2t}\) m
- 18
\(y=\dst{y_0\cos\omega_0 t+{v_0\over\omega_0}\sin\omega_0t;\; R={1\over\omega_0} \sqrt{(\omega_0y_0)^2+(v_0)^2}}\);
\(\cos\phi=\dst{y_0\omega_0\over \sqrt{(\omega_0y_0)^2+(v_0)^2}}\); \(\sin\phi=\dst{v_0\over \sqrt{(\omega_0y_0)^2+(v_0)^2}}\)
- 19
The object with the longer period weighs four times as much as the other.
- 20
\(T_2=\sqrt{2}T_1\), where \(T_1\) is the period of the smaller object.
- 21
\(k_1=9k_2\), where \(k_1\) is the spring constant of the system with the shorter period.
6.2 Spring Problems II
- 1
\(y=\dst{{e^{-2t}\over2}(3\cos2t-\sin2t)\mbox{ ft}; \; \sqrt{{5\over2}} e^{-2t}}\) ft
- 2
\(y=\dst{-e^{-t}\left(3\cos3t+{1\over3}\sin 3t\right)\mbox{ ft}\; {\sqrt{82}\over3}e^{-t}}\) ft
- 3
\(y=\dst{e^{-16t}\left({1\over4}+10t\right)}\) ft
- 4
\(y=\dst{-{e^{-3t}\over4}(5\cos t+63\sin t)}\) ft
- 5
\(0\le c<8\) lb-sec/ft
- 6
\(y=\dst{{1\over2}e^{-3t}\left(\cos\sqrt{91}t+{11\over \sqrt{91}}\sin\sqrt{91}t\right)}\) ft
- 7
\(y=-\dst{{e^{-4t}\over3} (2+8t)}\) ft
- 8
\(y=\dst{e^{-10t}\left(9\cos4\sqrt{6}t+{45\over 2\sqrt{6}}\sin4\sqrt{6} t\right)}\) cm
- 9
\(\dst{y=e^{-3t/2}\left({3\over2}\cos {\sqrt{41}\over2}t +{9\over2\sqrt{41}}\sin{\sqrt{41}\over2}t\right)}\) ft
- 10
\(y=\dst{e^{-{3\over2}t}\left({1\over2}\cos {\sqrt{119}\over2} t-{9\over2\sqrt{119}}\sin{\sqrt{119}\over2}t\right)}\) ft
- 11
\(\dst{y=e^{-8t}\left({1\over4}\cos8\sqrt{2}t-{1 \over4\sqrt{2}}\sin8\sqrt{2}t\right)}\) ft
- 12
\(\dst{y=e^{-t}\left(-{1\over3}\cos3\sqrt{11} t+ {14\over9\sqrt{11}}\sin3\sqrt{11}t\right)}\) ft
- 13
\(\dst{y_p={22\over61}\cos2t+{2\over61}\sin2t}\) ft
- 14
\(y=-\dst{2\over3}(e^{-8t}-2e^{-4t})\)
- 15
\(y=\dst{e^{-2t}\left({1\over10}\cos 4t-{1\over5}\sin4t\right)}\) m
- 16
\(y=e^{-3t}(10\cos t-70\sin t)\) cm
- 17
\(\dst{y_p=-{2\over15}\cos3t+{1\over15}\sin3t}\) ft
- 18
\(y_p=\dst{{11\over100}\cos4t+{27\over100}\sin4t}\) cm
- 19
\(\dst{y_p={42\over73}\cos t+{39\over73}\sin t}\) ft
- 20
\(y=\dst{-{1\over2}\cos2t+{1\over4}\sin2t}\) m
- 21
\(\dst{y_p={1\over c\omega_0}(-\beta\cos\omega_0t+ \alpha\sin\omega_0t)}\)
- 24
\(y=\dst{e^{-ct/2m} \left(y_0\cos\omega_1t+{1\over\omega_1}(v_0+{cy_0\over 2m})\sin\omega_1t\right)}\)
- 25
\(y=\dst{{r_2y_0-v_0\over r_2-r_1} e^{r_1t}+{v_0-r_1y_0\over r_2-r_1}e^{r_2t}}\)
- 26
\(y=e^{r_1t}\dst{\left(y_0+(v_0-r_1y_0)t\right)}\)
6.3 The $RLC$ Circuit
- 1
\(I=e^{-15t}\dst{\left(2\cos5\sqrt{15}t- {6\over\sqrt{31}}\sin5\sqrt{31}t\right)}\)
- 2
\(I=e^{-20t}(2\cos40t-101\sin40t)\)
- 3
\(I=-\dst{200\over3}e^{-10t}\sin30t\)
- 4
\(I=-10e^{-30t}(\cos40t+18\sin40t)\)
- 5
\(I=-e^{-40t}(2\cos30t-86\sin30t)\)
- 6
\(I_p=\dst-{1\over3}(\cos10t+2\sin10t)\)
- 7
\(I_p=\dst{20\over37}(\cos25t-6\sin25t)\)
- 8
\(I_p= \dst{3\over13}(8\cos50t-\sin50t)\)
- 9
\(I_p=\dst{20\over123}(17\sin100t-11\cos100t)\)
- 10
\(I_p=-\dst{45\over52}(\cos30t+8\sin30t)\)
- 12
\(\omega_0=1/\sqrt{LC}\) maximum amplitude \(=\sqrt{U^2+V^2}/R\)
6.4 Motion Under a Central Force
- 1
If \(e=1\), then \(Y^2=\rho(\rho-2X)\); if \(e\ne1\) \(\dst{\left(X+{e\rho\over1-e^2}\right)^2}+\dst{Y^2\over1-e^2}= \dst{\rho^2\over(1-e^2)^2}\) if ;
\(e<1\) let \(X_0=-\dst{e\rho\over1-e^2}\), \(a=\dst{\rho\over1-e^2}\), \(b=\dst{\rho\over\sqrt{1-e^2}}\).
- 2
Let \(h=r_0^2\theta_0'\); then \(\rho=\dst{h^2\over k}\), \(e=\dst{\left[\left({\rho\over r_0}-1\right)^2+\left(\rho r_0'\over h\right)^2\right]^{1/2}}\). If \(e=0\), then
\( \theta_0\) is undefined, but also irrelevant if \(e\ne0\) then \(\phi=\theta_0-\alpha\), where \(-\pi\le\alpha<\pi\), \(\cos\alpha=\dst{{1\over e}\left({\rho\over r_0}-1\right)}\) and \(\sin\alpha=\dst{\rho r_0'\over eh}\).
- 3
(a) \(e=\dst{\gamma_2-\gamma_1\over\gamma_1+\gamma_2}\) (b) \(r_0=R\gamma_1\), \(r_0'=0\), \(\theta_0\) arbitrary, \(\theta_0'=\dst{\left[2g\gamma_2\over R\gamma_1^3(\gamma_1+\gamma_2)\right]^{1/2}}\)
- 4
\(f(r)=-mh^2\dst\left({6c\over r^4}+{1\over r^3}\right)\)
- 5
\(f(r)=-\dst{mh^2(\gamma^2+1)\over r^3}\)
- 6
(a) \(\dst{d^2u\over d\theta^2}+\left(1-{k\over h^2}\right)u=0,\; u(\theta_0)=\dst{1\over r_0},\; \dst{du(\theta_0)\over d\theta}=-\dst{r_0'\over h}\). (b) with \(\gamma=\)
\(\left|1-\dst{k\over h^2}\right|^{1/2}\): (i) \(r=r_0\left( \cosh\gamma(\theta-\theta_0)-\dst{r_0r_0'\over\gamma h}\sinh\gamma(\theta-\theta_0)\right)^{-1}\) (ii) \(r=r_0\left(1 -\dst{r_0r_0'\over h}(\theta-\theta_0)\right)^{-1}\); (iii) \(r=r_0\left( \cos\gamma(\theta-\theta_0)-\dst{r_0r_0'\over\gamma h}\sin\gamma(\theta-\theta_0)\right)^{-1}\)