솔루션피아

$$ \begin{cases} T&=30\ut{s}\\ R&=4.0\ut{m}\\ \end{cases} $$ $$\ab{a}$$ $$ \begin{aligned} v&=\frac{2\pi R}{T}\\ &=\frac{4\pi}{15}\\ &\approx 0.8377580409572781\ut{m/s}\\ &\approx 0.84\ut{m/s}\\ \end{aligned} $$ $$\ab{b}$$ $$\Sigma F_R = \frac{mv^2}{R},$$ $$\mu F_N=\mu m g=\frac{mv^2}{R}$$ $$ \begin{aligned} \mu &=\frac{v^2}{gR}\\ &=\frac{4\pi^2 R}{g T^2}\\ &=\frac{4\pi^2}{225g}\\ &=0.0178919033..

$$ \begin{cases} R&=200\ut{m}\\ g&=9.80665\ut{m/s^2} \end{cases} $$ $$\Sigma F_R = \frac{mv^2}{R},$$ $$ \begin{aligned} \frac{mv^2}{R}&=mg\\ v&=\sqrt{gR}\\ &\approx 44.28690551393267\ut{m/s}\\ &\approx 4\times10\ut{m/s}\\ \end{aligned} $$

$$ \begin{cases} m_A&=5.0\ut{kg}\\ m_B&=3.0\ut{kg}\\ \mu&=0.50\\ \theta&=30\degree\\ \end{cases} $$ $$\Sigma \vec F = m\vec a,$$ $$ \begin{cases} \Sigma F_{Ax} &= m_A a\\ \Sigma F_{Ay} &= 0\\ \Sigma F_{Bx} &= m_B a\\ \Sigma F_{By} &= 0\\ \end{cases} $$ $$ \begin{cases} m_A a &= m_Ag\sin\theta-T\\ 0 &= N_A-m_Ag\cos\theta\\ m_B a &= T-\mu N_B\\ 0 &= N_B-m_Bg\\ \end{cases} $$ $$\ab{a}$$ $$ \begin{a..

$$ \begin{cases} m_1&=2.0\ut{kg}\\ m_2&=1.0\ut{kg}\\ F&=28\ut{N},\theta=35\degree\\ \mu&=0.28 \end{cases} $$ $$\Sigma \vec F = m\vec a,$$ $$ \begin{cases} \Sigma F_{1x} &= m_1 a\\ \Sigma F_{1y} &= 0\\ \Sigma F_{2x} &= m_2 a\\ \Sigma F_{2y} &= 0\\ \end{cases} $$ $$ \begin{cases} m_1 a &= T-\mu N_1\\ 0 &= N_1-m_1g\\ m_2 a &= F\cos\theta-T-\mu N_2\\ 0 &= N_2-F\sin\theta-m_2g\\ \end{cases} $$ $$ \be..

$$ \begin{cases} m&=12.0\ut{kg}\\ \mu&=0.380\\ g&=9.80665\ut{m/s^2} \end{cases} $$ $$\Sigma \vec F = 0 \Harr \Delta \vec v=0,$$ $$ \begin{cases} \Sigma F_x &= 0\\ \Sigma F_y &= 0\\ \end{cases} $$ $$ \begin{cases} 0 &= F\cos\theta-\mu N\\ 0 &= F\sin\theta+N-mg\\ \end{cases} $$ $$ \begin{aligned} F&=\frac{\mu \text{mg}}{\mu \sin\theta+\cos\theta} \end{aligned} $$ $$\ab{a}$$ $$\theta=0,$$ $$ \begin..

$$ \begin{cases} a&=0.60\ut{m/s^2}\\ \theta&=35\degree\\g&=9.80665\ut{m/s^2} \end{cases} $$ $$\Sigma \vec F = m\vec a,$$ $$ \begin{cases} \Sigma F_x &= m a_x\\ \Sigma F_y &= 0\\ \end{cases} $$ $$ \begin{cases} m a &= mg\sin\theta-\mu N\\ 0 &= N-mg\cos\theta\\ \end{cases} $$ $$ \begin{aligned} \mu&=\tan\theta-\frac{a}{g\cos\theta}\\ &\approx 0.6255169197766185\\ &\approx 0.63\\ \end{aligned} $$

$$ \begin{cases} m&=6.70\ut{kg}\\ \mu&=0.850\\ \theta&=65.0\degree\\ g&=9.80665\ut{m/s^2} \end{cases} $$ $$\Sigma \vec F = 0 \Harr \Delta \vec v=0,$$ $$ \begin{cases} \Sigma F_x &= 0\\ \Sigma F_y &= 0\\ \end{cases} $$ $$ \begin{cases} 0 &= F\cos\theta-\mu N\\ 0 &= F\sin\theta-mg-N\\ \end{cases} $$ $$0 = F\cos\theta-\mu (F\sin\theta-mg)$$ $$ \begin{aligned} F&=\frac{\mu mg}{\mu\sin\theta-\cos\the..

$$ \begin{cases} m_1&=2.0\ut{kg}\\ m_2&=4.0\ut{kg}\\ \theta&=40\degree\\ \mu&=0.25\\g&=9.80665\ut{m/s^2} \end{cases} $$ $$ \Sigma \vec F = m\vec a, $$ $$ \begin{cases} \Sigma F_{1x} &= m_1 a_x\\ \Sigma F_{1y} &= 0\\ \Sigma F_{2x} &= m_2 a_x\\ \Sigma F_{2y} &= 0\\ \end{cases} $$ $$ \begin{cases} m_1 a&=T-\mu N_1\\ 0&=N_1-m_1g\\ m_2 a&=m_2g\sin\theta-T\\ 0&=N_2-m_2g\cos\theta\\ \end{cases} $$ $$ \..

$$ \begin{cases} R&=520\ut{m}\\ m&=51.0\ut{kg}\\ F_x&=165\ut{N}\\ F_y&=500\ut{N} \end{cases} $$ $$\ab{a}$$ $$ \begin{aligned} \Sigma \vec F&=\Sigma F_R=F_x\\ &=165\ut{N} \end{aligned} $$ $$\ab{b}$$ $$\Sigma F_R=\frac{mv^2}{R}$$ $$ \begin{aligned} v&=\sqrt\frac{R F_x}{m}\\ &=10\sqrt{\frac{286}{17}}\ut{m/s}\\ &\approx41.01649596414192\ut{m/s}\\ &\approx41.0\ut{m/s}\\ \end{aligned} $$

$$ \begin{cases} R&=3.00\ut{cm}\\ m&=4.00\ut{kg}\\ C&=1.80\\ \rho&=1.20\ut{kg/m^3}\\ g&=9.80665\ut{m/s^2} \end{cases} $$ $$\Sigma \vec F = 0 \Harr \Delta \vec v=0,$$ $$mg=\frac{1}{2}C\rho Av^2$$ $$ \begin{aligned} v&=\sqrt{\frac{2mg}{C\rho A}}\\ &=\frac{1000}{9}\sqrt{\frac{g}{3\pi}}\\ &\approx 113.33975372780418\ut{m/s}\\ &\approx 113\ut{m/s} \end{aligned} $$