for which the garbage will fall within the can. If the range is large, the shot is easy; if the range is small, the target is hard to hit.

Transcription

1 EN40: Dynamics an Vibrations Homework : Soling equations of motion for particles Due Friay Feb School of Engineering Brown Uniersity 1. Suppose that you wish to throw a piece of garbage into a garbage can. Is it better to use an unerarm or oerarm throw? A etaile analysis of this important question has been publishe by Professor Mahaean an his group at Harar, 0.5m (who is eiently putting his MacArthur fellowship to goo use). To show that first-year stuents at Brown are as smart as faculty at Harar, you will work through a simplifie ersion of their calculations. To keep things simple, we will make 1m j the following assumptions: You throw by rotating your arm at the i elbow, in a seate position. Your elbow has the same height for both uner- an oerarm.75m throws 0.5m You throw the projectile with a spee of 5 m/s, for both oer- an uner-arm throws. Releant imensions are shown in the figure. Your garbage will go into the basket only if you release the projectile at the right point. Specifically, there will be some range of release angles 1 for which the garbage will fall within the can. If the range is large, the shot is easy; if the range is small, the target is har to hit. 1.1 Start by analyzing an oerarm throw. Assume that the projectile is release with your arm at angle to the ertical. Fin a formula for the position ector of the projectile at time t after release, using the coorinate system shown in the figure. 1. Hence, calculate an expression for the time that the projectile reaches height h=0, in terms of 1. Derie an equation for the release angle that will just hit the outsie of the garbage can (on t try to sole the equation by han). Using MAPLE, fin two possible release angles that satisfy the equation. (if you woul like to stop MAPLE giing complex alue solutions, you can type with(realdomain) at the start of your MAPLE file. 1.4 Derie an equation for the release angle that will just hit the insie of the garbage can, an once again, use MAPLE to fin the two possible release angles satisfying the equation. 1.5 Hence, calculate 1 for the oerarm throw (you will get two ranges, of course use the larger) 1.6 Repeat for an unerarm throw (look for a quick way to o this you can just make a couple of small changes to your MAPLE file to get the answers you nee). Which throwing style is preferable?

2 . The figure illustrates a simple screening test for anemia (known as the CuSO4 test see, e.g. Rumann Hanbook of bloo banking an transfusion meicine ). The x technician rops a bloo specimen into a CuSO4 solution with carefully controlle concentration, an obseres the rop for 15sec. If the rop floats, or sinks less than a critical istance, the test inicates that the patient may hae anemia. The goal of this problem is to etermine the require concentration of the solution. Assume that A bloo cell can be iealize as a sphere with raius R an mass ensity The CuSO4 solution has mass ensity s an iscosity The rag force on a sphere raius R moing through a iscous flui with spee is 6 R.1 Draw a free boy iagram showing the forces acting on one of the bloo cells within the bloo roplet.. Write own Newton s law of motion relating the epth x of the bloo cell to releant parameters. Show that x satisfies the ifferential equation x 9 x ( / 1) 0 s g R. Assume that a representatie bloo cell starts at x=0 an has zero spee at time t=0. Hence, calculate expressions for the spee, an epth, of the bloo cell in terms of time an other releant parameters. Use the MAPLE sole function to sole the ifferential equation..4 Use the following alues for parameters: mass ensity of normal bloo cells 1.19gram cm, mass ensity of CuSO4 solution s ( c) / ( c) gram / cm, where c is the molar concentration of CuSO4 in the solution (the number of mols of CuS04 pentahyrate iie by the 6 Bloo CuSO 4 number of mols of water), iscosity 10 Ns m, raius of a bloo cell R.5 m. Determine the concentration c that will result in bloo sinking by 4cm uring the 15 secon test perio.

3 . The figure shows a rocket, which may be iealize as a particle with mass m, an which traels close to the earth s surface. It is launche from the origin with initial elocity ector V0 Vxi Vy j Vzk. The projectile is subjecte to the force of graity (acting in the negatie k irection), a rag force 1 FD CD AV where V x y z is the magnitue of the rocket s elocity, xi y j zk is the projectile elocity, CD is the rag coefficient, an is the air ensity. In aition, a thrust force FT F0 / V is exerte on the rocket by its motor (the thrust acts in the irection of motion of the rocket)..1 The motion of the system will be escribe using the (x,y,z) coorinates of the particle. Write own the acceleration ector in terms of time eriaties of these ariables.. Write own the ector equation of motion for the particle (F=ma).. Show that the equation of motion can be expresse in MATLAB form as x x y y z z x c1vx cx / V y c1vy cy / V z g c1vz cz / V an gie formulas for the constants c 1 an c in terms of releant parameters..4 Moify the MATLAB coe iscusse in class (or the online notes, or the MATLAB tutorial, or last year s HW) to calculate an plot the trajectory. A an Eent function to your coe that will stop the calculation when the rocket reaches the groun (THERE IS NO NEED TO SUBMIT A SOLUTION TO THIS PROBLEM)..5 Finally, run simulations with parameters representing a (large) moel rocket (m=1kg, CD 0.1, A=0.015m, air ensity 1.0 kg/m, F0 0N ) launche in the (x-z) plane at m/s at an angle of egrees to the ertical. Plot the trajectory of the missile, an etermine the time of flight, an the total horizontal istance traele..6 Optional for extra creit. Suppose the rocket motor has a 5sec burn (i.e. the thrust is zero after 5sec). Fin the initial launch angle that maximizes the horizontal range of the rocket, an fin the corresponing range. Also, etermine the maximum height reache by the rocke uring its flight. NOTE oe45 is not accurate enough for this problem you shoul use oe11 instea. Also, set the Reltol parameter to or better.

4 4. A Paul Trap is one of seeral eices that use specially shape electrostatic an electromagnetic fiels to trap charge particles (or ions ) (see, e.g. Major et al (005) for more information). It has a number of applications: for example, as a mass spectrometer (use in chemical an biochemical analysis); or as a component in experimental quantum computers. The ions insie a particle trap are continuously moing, an follow ery complicate orbits. The goal of this problem is to analyze the motion of a single charge particle insie a Paul trap. 4.1 The motion of the ion will be escribe by the components of its position ector (x,y,z). Write own the elocity an acceleration of the ion in terms of time eriaties of these ariables. 4. The ion has mass m, an is subjecte to a force F QE where Q is its charge, an E0 (1 cos t E ) xi yj zk is a time epenent electric fiel ector. Here, E 0,, are constants that specify the magnitue an geometry of the electric an magnetic fiels. Use Newton s law to show that the components of the position ector of the ion satisfies the following equations of motion where x (1 cos( t)) x y (1 cos( t)) y z (1 cos( t)) z is a parameter (which has units of frequency). QE 0 m 4. Re-write the equations of motion as 6 first-orer ifferential equations that can be integrate using MATLAB. 4.4 Write a MATLAB script that will sole the equations of motion to etermine x,y,z, x, y, z as a function of time. You nee not submit a solution to this problem. 4.5 An ion will be trappe if the frequency of the electric fiel is relate to correctly, an if the amplitue of the oscillation lies in the correct range. Demonstrate this behaior by running computations with the following parameters (the units are arbitrary in actual esigns the frequencies are ery high; typically raio frequencies). (a) x z 0.1 y x z y (b) x z 0.1 y 0 x z 0 y (c) x z 0.1 y 0 x z 0 y () x z 0.1 y 0 x z 0 y Run the tests for a time interal of 60 units. Han in a graph showing the preicte trajectory for each case.

5 5. The figure shows a schematic of the parametrically excite inerse penulum iscusse on the first ay of class. The piot point moes ertically with a isplacement x X 0 sin t. The goal of this problem is to show that at an appropriate frequency the inerte penulum is stable. 5.1 Write own the position ector of the mass m. Hence, calculate an expression for the acceleration of the mass, in terms of, X0, L as well as an its time eriaties. 5. Draw a free boy iagram showing the forces acting on the mass m j m L Actuator x i 5. Write own Newton s law of motion of the mass, an hence show that the angle satisfies the equation g X (1 0 sin t)sin 0 L g 5.4 Arrange the equation of motion for into a ector form that MATLAB can sole. 5.5 Write a MATLAB program to calculate an plot the angle as a function of time. Show the angle in egrees. Use a relatie tolerance of in the ODE soler (You on t nee to submit a solution to this problem) 5.6 Plot graphs of as a function of time for the following parameters. Run each simulation for 0 sec, an initial conitions 5eg 0 (a) 60 (0cycles / sec) L 0.5m X m (b) 50 (5cycles / sec) L 0.5m X m (c) 60 (0cycles / sec) L 0.5m X m () 7 (6.5cycles / sec) L 0.5m X m

6 6. OPTIONAL: FOR EXTRA CREDIT The figure shows a conceptual esign for a space eleator which (if materials scientists an electrical engineers are able to achiee seeral technological miracles) offers a ery low-cost approach to launching payloas into orbit. It consists of a satellite with mass M, which lies in the equatorial plane an is in geostationary orbit. The satellite is tethere to the earth by a cable. A crawler with mass m ries up an own this cable transporting freight an passengers from the earth s surface to orbit. The goal of this problem is to analyze the motion of this system. For simplicity, we will Assume that the system remains in the equatorial plane, so that the position of the satellite an crawler can be escribe by their coorinates in a fixe Cartesian basis r1 x1i y1j, r xi yj Neglect the mass of the cables Iealize the cables below an aboe the crawler as a combination of a spring an amper, which exert forces T1 k ( L1 a1 ) 1 ( L1 a1 ) T k ( L a ) ( L a ) to the objects attache to their ens, where L1, L are the stretche lengths of the cables, a1, a are the un-stretche cable lengths, an k1, k an 1, are cable stiffnesses an amping coefficients. 6.1 The point X where the cable is attache to the surface of the earth moes as the earth rotates. Assume that at time t=0 this point has position ector rx Ri, where R is the raius of the earth. Write own (a) the position ector of X as a function of time, an (b) the elocity ector terms of R an the angular spee of the earth X of point X, in 6. Write own formulas for (a) the cable lengths; (b) unit ectors parallel to each cable; (c) unit ectors parallel to the graitational forces acting on the crawler an satellite in terms of ( x1, y 1), ( x, y ). Also, fin expressions for L 1 / an L / in terms of ( x1, y 1), ( x, y) an their time eriaties. 6. Write own the acceleration ectors for the crawler an the satellite, in terms of eriaties of ( x1, y 1), ( x, y) 6.4 Draw two free boy iagrams showing (a) the forces acting on the satellite, an (b) the forces acting on the crawler.

7 6.5 Show that the equations of motion for the crawler an satellite can be expresse as x1 x1 y y1 1 x x y y x1 x1 / r1 T1 ( R cos t x1 ) / ml1 T ( x x1 ) / ml y1 y1 / r1 T1 ( R sin t y1 ) / ml1 T ( y y1 ) / ml x x / r T ( x1 x ) / ML y y / r T ( y1 y ) / ML where is the prouct of the earth s mass an the graitational constant an r1 x1 y1 r x y 6.6 Write a MATLAB script that will calculate the position an elocity of the satellite an crawler as a function of time. Use the following alues for parameters in the problem 5 1 The graitational parameter km s The earth s raius R is 647km The earth s angular elocity is / 400 ra/s The cables hae constant stiffness k1 k 1 kn / km an amping 1 1 kns / km The satellite has initial position an elocity r i km jkm/s The crawler has initial position an elocity r i km jkm/s As the crawler climbs the cable, the un-stretche lengths of the two cables ary with time. Take the un-stretche lengths (in km) to be ( t / T sin t / T ) t T a1 4 t T a 586 a1 where T is the time of ascent. There is no nee to submit a solution to this problem. 6.7 Plot a graph showing the trajectory (x--y) of the crawler an satellite as the crawler ascens the cable. Plot both trajectories on the same graph, but show them in ifferent colors. Try the following parameters, an run all calculations for a total time of 8 ays. M kg m 500kg T 5ays M 50000kg m 500kg T 5ays M kg m 500kg T 1ay 6.8 Plot a graphs showing the tension in each of the cables as a function of time.