Question

pls answer all, i will rate and thumbs up.

studying hard for my exam.

1. For a particle moving in the r-y plane, the position can be written in polar coordinates as r rf, where r= V be written as + yj? and f = cos 0i + sin0j. Show that the velocity can v iTr where the unit vector is orthogonal to f. [4 marks 2. A cylinder of mass M and radius R is rolling along a surface without slipping at a speed v. Show that the total kinetic energy of the cylinder is TM2 [4 marks 3. A straight stick is standing on end on a frictionless surface. Sketch the movement of the stick as it falls down, indicating the motion of its centre of mass 4 marks 4. The pendulum clock was invented by Huygens in 1656. A few years later, he observed that if two clocks with the same length pendulum are mounted on a common wall, they tend to synchronise exactly out of phase. Why does this happen? [4 marks 5. Argue why the equilibrium vapour pressure, that is, the pressure exerted by a vapour in thermodynamic equilibrium with its condensed phase (solid or liquid), must be a function of temperature [4 marks

thanks so much

Answer 1

$\dpi{80} 1.\;\begin{align*} \;\;\vec{r}&= r\;\hat{r}\\ \dfrac{\vec{r}}{dt}&=\dfrac{\partial r}{\partial t}\;\hat{r}\;+\;r\;\dfrac{\partial \hat{r}}{\partial t}\\ &= \dot{r}\;\hat{r}\;+\;r\;\dfrac{\partial \hat{r}}{\partial t}\\ \dfrac{\partial \hat{r}}{\partial t}&=\dfrac{\partial \hat{r}}{\partial \theta}\;\dfrac{\partial \theta}{\partial t}\\ &= (- sin\theta \hat{i} + cos\theta \hat{j})\;\;\dot{\theta}\\ &=\dot{\theta}\;\hat{\theta}\\\\ \;\textbf{Total K.E of rolling object}\;&=\;\dfrac{1}{2}\;MV^2\;(1+\dfrac{K^2}{R^2})\\\\ &\textbf{R be the radius and K be the radius of gyration}\\ \textbf{For a cylinder rolling },& M.I = \dfrac{1}{2}\;MR^2\\ MK^2&=MR^2/2\\ \dfrac{K^2}{R^2}&=\dfrac{1}{2} \\So \; K.E &= \dfrac{1}{2}\;MV^2\;(1+1/2)\\ &=\dfrac{3}{4}\;MV^2\\\\ \end{align*}$

3. Since there are no external in horizontal direction, path will be a straight line verically downwards

4.  Bifurcations in non linear dynamics of weakly coupled oscillators

5. As number increases, according to Kinetic theory , the pressure must also increase

Answer 2

$\dpi{80} 1.\;\begin{align*} \;\;\vec{r}&= r\;\hat{r}\\ \dfrac{\vec{r}}{dt}&=\dfrac{\partial r}{\partial t}\;\hat{r}\;+\;r\;\dfrac{\partial \hat{r}}{\partial t}\\ &= \dot{r}\;\hat{r}\;+\;r\;\dfrac{\partial \hat{r}}{\partial t}\\ \dfrac{\partial \hat{r}}{\partial t}&=\dfrac{\partial \hat{r}}{\partial \theta}\;\dfrac{\partial \theta}{\partial t}\\ &= (- sin\theta \hat{i} + cos\theta \hat{j})\;\;\dot{\theta}\\ &=\dot{\theta}\;\hat{\theta}\\\\ \;\textbf{Total K.E of rolling object}\;&=\;\dfrac{1}{2}\;MV^2\;(1+\dfrac{K^2}{R^2})\\\\ &\textbf{R be the radius and K be the radius of gyration}\\ \textbf{For a cylinder rolling },& M.I = \dfrac{1}{2}\;MR^2\\ MK^2&=MR^2/2\\ \dfrac{K^2}{R^2}&=\dfrac{1}{2} \\So \; K.E &= \dfrac{1}{2}\;MV^2\;(1+1/2)\\ &=\dfrac{3}{4}\;MV^2\\\\ \end{align*}$

3. Since there are no external in horizontal direction, path will be a straight line verically downwards

4.  Bifurcations in non linear dynamics of weakly coupled oscillators

5. As number increases, according to Kinetic theory , the pressure must also increase