EN40: Dynamics and Vibrations. Midterm Examination Tuesday March

Transcription

1 EN4: Dynaics and Vibrations Midter Exaination Tuesday March 8 16 School of Engineering Brown University NAME: General Instructions No collaboration of any kind is peritted on this exaination. You ay bring double sided pages of reference notes. No other aterial ay be consulted Write all your solutions in the space provided. No sheets should be added to the exa. Make diagras and sketches as clear as possible, and show all your derivations clearly. Incoplete solutions will receive only partial credit, even if the answer is correct. If you find you are unable to coplete part of a question, proceed to the next part. Please initial the stateent below to show that you have read it `By affixing y nae to this paper, I affir that I have executed the exaination in accordance with the Acadeic Honor Code of Brown University. PEASE WRITE YOUR NAME ABOVE ASO! 1 (19 points). (11 points) 3. (1 points) TOTA (4 points)

2 1. The figure shows an inertial crawler that will spontaneously translate to the right over a vibrating surface. Assue that The device and surface are both at rest at tie t=. For the tie interval <t<t (where T is a constant) the surface has velocity + Vi < t < T / v = Vi T / < t < T V is sufficiently large that slip occurs at A throughout Center of ass the interval <t<t. The contact at A has a friction coefficient µ. The wheel at B is assless and rolls freely (so the contact at B is subjected only to a noral force). The center of ass of the device is idway between A and B and a height h above the surface. A h Base otion j B i 1. Draw a free body diagra showing the forces acting on the crawler during the period < t < T / on the figure provided below. Note that during this phase of the otion the surface oves to the right, and is oving faster than the crawler. j i A h B [3 POINTS]. Write down Newton s law and the equation of rotational otion (i.e. oents) for < t < T / [3 POINTS]

3 3. Hence, show that the acceleration of the crawler for < t < T / is given by µ g a = i µ ( h/ ) [3 POINTS] 4. Repeat steps 1-3 for the period T /< t < T (note that during this phase of the otion the surface oves to the left) to show that during this tie interval µ g a = i + µ ( h/ ) [4 POINTS] 3

4 5. Use the results of (3) and (4) to find a forula for the distance that the crawler oves during the tie period <t<t. Assue that the crawler is at rest at tie t=. [4 POINTS] 6. Find a forula for the iniu value of V for slip to occur at the contact point A between the crawler and the surface in the tie interval <t<t. 4

5 . The figure shows a charged particle with ass in a static Kingdon trap that uses an electric field to confine the particle within a cylinder. The electric field subjects the particle to a radial force F F= e r r where F is a constant (which depends on the electric field in the cylinder). At tie t= the particle is located at position θ =,r = R and has velocity vector v= Ve θ j e θ ion r Wire Cylinder θ i e r.1 Write down the forula for the acceleration of the particle in polar coordinates, in ters of tie derivatives of r, θ. (Do not assue circular otion).. Hence, write down Newton s law F=a for the particle, using polar coordinates. [1 POINT] 5

6 .3 Write down the angular oentu vector of the particle at tie t=, in ters of V, and R [1 POINT].4 Use angular oentu, or otherwise, show that θ and r are related by dθ VR = dt r 6

7 .5 Use the answers to. and.4 to show that the coordinate r satisfies the differential equation ( VR) F r + = 3 d r dt r [3 POINTS].6 Rearrange the equation of otion for r in part 3.3 into a for that MATAB can solve. 7

8 3. The figure shows a device that is intended to detect an ipulse. It consists of a casing, ass M, a proof ass,, and a spring with stiffness k and unstretched length. If the casing is subjected to an ipulse that exceeds a critical agnitude, the sall ass will flip fro its initial position to the right of the pivot to a new stable position to the left of the pivot at A. The goal of this proble is to calculate the critical value of ipulse for which this will occur. 3.1 Suppose that at tie t= the syste is at rest and the spring is un-stretched. The casing is then subjected to a horizontal ipulse I. Write down a forula for the speed of the casing just after the ipulse. Note that the spring exerts no force on either the casing or the ass during the ipulse. Ipulse I Just after ipulse / y(t) x() Α V k, k, / x(t) Α Casing Mass M M y(t) V 1 Spring at shortest length / Α k, M x(t) [1 POINT] y(t) 3. Find expressions for the total linear oentu and total kinetic energy of the syste just after the ipulse, in ters of I and the ass M of the casing. 8

9 3.3 Consider the syste at the instant when the spring reaches its shortest length (assue x> ). Using energy and/or oentu conservation show that at this instant x= I km ( M ) + 4 (You can find the spring length using Pythagoras theore. Note that is the un-stretched length of the spring) 3.4 Hence, find a forula for the critical value of I that will flip the ass past x=. [5 POINTS] 9