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pupiloslash
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Hello friends .. I did a physics exam last week and had many doubts on questions 1 - d) e) and 2 - b) c) ... I tried to solve the test at home but I could not .. if someone can give a help I would greatly appreciate it! items d) and e) the question is a damped harmonic oscillations on the fluid (air) and the second question is about dynamics of spin ... the teacher is based on the book Moyses, thank you already!
1) Consider a frontal collision between two particles, A and B respectively mass M and m moving in straight line along the same trajectory (axis x), without friction. The particle A is at rest x = 0 and particle B approaches A with speed -V <0 coming from x> 0 . Where M>> m , after an elastic collision (energy saving) it is observed that the particle B moves away from A with the speed v> 0 . The final speed of A can be considered null with respect to v .
(b) Determine the final momentum of A , [URL]http://latex.codecogs.com/gif.latex?p_{A}^{f}%20=%20\Delta%20p_{A}[/URL]
(c) Describe the impact (ie, determine the velocities of A and B before and after the collision) when B is initially at rest x0 and A approaches coming from x <x0 with the speed v . [Note that this is just a change of reference with respect to (b)]. Determine [URL]http://latex.codecogs.com/gif.latex?\Delta%20p_{A}[/URL] this case.
(d) In a hypothetical situation, A moves with velocity v in a uniform density of gas molecules of mass m , all initially at rest, occurring on average one collision every time interval [URL]http://latex.codecogs.com/gif.latex?\delta%20t.[/URL] Each collision produces the same variation of momentum of A calculated in (c). Determine the effective force on the weight M . Consider a time interval to measure the effective force [URL]http://latex.codecogs.com/gif.latex?\Delta%20t\gg%20\delta%20t[/URL]
(e) The particle A in the gas described in (d) is attached to the end of a spring with constant k and which oscillates along the axis x . Write the equation of motion. What behavior do you expect to find the amplitude of the movement for [URL]http://latex.codecogs.com/gif.latex?2\pi%20\sqrt{(M/k)}[/URL]
2) A bar table has a top spinning metal ring radius R and mass M . A bullet of mass [URL]http://latex.codecogs.com/gif.latex?m%20\ll%20M[/URL] hits the table with speed [URL]http://latex.codecogs.com/gif.latex?\vec{v}%20=%20v%20\widehat{\theta}[/URL] (tangential) on board in [URL]http://latex.codecogs.com/gif.latex?\vec{r}%20=%20R%20\widehat{r}.[/URL] We use polar coordinates with respect to the center of the table that can rotate without friction. The moment of inertia of the table in the center is [URL]http://latex.codecogs.com/gif.latex?I%20=%20MR^{2}/2.[/URL] Determine the angular velocity of the final table in the following cases:
(a) The ball gets stuck on the edge of the table.
(b) The bullet grazed almost using the same trajectory with velocity [URL]http://latex.codecogs.com/gif.latex?{\vec{v}}'%20=%20\vec{v}/2.[/URL]
(c) Determine, in both cases (a) and (b), the ratio between final and initial kinetic energy. Remember that [URL]http://latex.codecogs.com/gif.latex?\frac{m}{M}%20\ll%201.[/URL]
1) Consider a frontal collision between two particles, A and B respectively mass M and m moving in straight line along the same trajectory (axis x), without friction. The particle A is at rest x = 0 and particle B approaches A with speed -V <0 coming from x> 0 . Where M>> m , after an elastic collision (energy saving) it is observed that the particle B moves away from A with the speed v> 0 . The final speed of A can be considered null with respect to v .
(b) Determine the final momentum of A , [URL]http://latex.codecogs.com/gif.latex?p_{A}^{f}%20=%20\Delta%20p_{A}[/URL]
(c) Describe the impact (ie, determine the velocities of A and B before and after the collision) when B is initially at rest x0 and A approaches coming from x <x0 with the speed v . [Note that this is just a change of reference with respect to (b)]. Determine [URL]http://latex.codecogs.com/gif.latex?\Delta%20p_{A}[/URL] this case.
(d) In a hypothetical situation, A moves with velocity v in a uniform density of gas molecules of mass m , all initially at rest, occurring on average one collision every time interval [URL]http://latex.codecogs.com/gif.latex?\delta%20t.[/URL] Each collision produces the same variation of momentum of A calculated in (c). Determine the effective force on the weight M . Consider a time interval to measure the effective force [URL]http://latex.codecogs.com/gif.latex?\Delta%20t\gg%20\delta%20t[/URL]
(e) The particle A in the gas described in (d) is attached to the end of a spring with constant k and which oscillates along the axis x . Write the equation of motion. What behavior do you expect to find the amplitude of the movement for [URL]http://latex.codecogs.com/gif.latex?2\pi%20\sqrt{(M/k)}[/URL]
2) A bar table has a top spinning metal ring radius R and mass M . A bullet of mass [URL]http://latex.codecogs.com/gif.latex?m%20\ll%20M[/URL] hits the table with speed [URL]http://latex.codecogs.com/gif.latex?\vec{v}%20=%20v%20\widehat{\theta}[/URL] (tangential) on board in [URL]http://latex.codecogs.com/gif.latex?\vec{r}%20=%20R%20\widehat{r}.[/URL] We use polar coordinates with respect to the center of the table that can rotate without friction. The moment of inertia of the table in the center is [URL]http://latex.codecogs.com/gif.latex?I%20=%20MR^{2}/2.[/URL] Determine the angular velocity of the final table in the following cases:
(a) The ball gets stuck on the edge of the table.
(b) The bullet grazed almost using the same trajectory with velocity [URL]http://latex.codecogs.com/gif.latex?{\vec{v}}'%20=%20\vec{v}/2.[/URL]
(c) Determine, in both cases (a) and (b), the ratio between final and initial kinetic energy. Remember that [URL]http://latex.codecogs.com/gif.latex?\frac{m}{M}%20\ll%201.[/URL]
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