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1 EN40: Dynamcs and Vbratons Mdterm Examnaton Thursday March Dvson of Engneerng rown Unversty NME: Isaac Newton General Instructons No collaboraton of any knd s permtted on ths examnaton. You may brng 3 double sded pages of reference notes. No other materal may be consulted Wrte all your solutons n the space provded. No sheets should be added to the exam. Make dagrams and sketches as clear as possble, and show all your dervatons clearly. Incomplete solutons wll receve only partal credt, even f the answer s correct. If you fnd you are unable to complete part of a queston, proceed to the next part. Please ntal the statement below to show that you have read t `y affxng my name to ths paper, I affrm that I have executed the examnaton n accordance wth the cademc Honor Code of rown Unversty. PLESE WRITE YOUR NME OVE LSO! IN 1 (3 ponts) (4 ponts) 3. (8 ponts) 4. (7 ponts) 5. (8 ponts) TOTL (30 ponts)

2 a a a (a) V (b) V (c) V 1. The fgure shows a car that travels along a crcular road. In Fg 1(a), the car travels at constant speed In Fg 1(a), the drver s brakng, and the car s speed s decreasng In Fg 1(c), the drver has her foot on the gas and the car s speed s ncreasng. Draw an arrow on each of fgures (a), (b), (c) to show the approxmate drecton of the car s acceleraton vector. [1 POINT ECH]. The fgure shows a vbraton measurement from a velocty transducer. The vbraton may be assumed to be harmonc. Estmate (a) The perod of oscllaton 4 cycles n 0.1sec gves T=0.1/4=0.05s (b) The angular frequency of oscllaton ngular frequency s 80 rad / s T (c) The ampltude of the velocty 5 cm/s (d) The ampltude of the dsplacement. X V / 5 / (80 ) 1/ (16 ) cm 0 0 [1 POINT ECH]

3 3. The goal of ths problem s to estmate the shortest stoppng dstance for a bcycle durng rear-wheel brakng. ssume that t tme t=0 the bcycle has velocty V. t ths nstant, the rder brakes hard enough to lock the rear wheel, causng t to skd. The coeffcent of frcton between the rear wheel and the ground s denoted by The front wheel rolls freely. r resstance may be neglected. h d L 3.1 Draw the forces actng on the bcycle and rder, usng the fgure shown. The bcycle and rder together may be dealzed as a partcle on a massless frame. T mg N N [ POINTS] 3. Wrte down Newton s law of moton F=ma and the moment balance equaton M=0 about the center of mass, expressng your answer as components n the bass shown. T ( N N mg) ma N ( L d) N d T h k Hence, calculate an expresson for the acceleraton of the bcycle n terms of, g, d and h. [ POINTS] The frcton law gves T N. The moment equaton and vertcal component of Newtons law are N N mg 0 Elmnate N : N ( L d) N ( d h) 0 ( L d) N N ( d h) mg( L d) 0 N mg( L d) / ( L h) Fnally the component of F=ma shows that a T / m N / m mg( L d) / ( L h) [ POINTS] 3

4 3.4 Deduce a formula for the stoppng dstance n terms of V,, g, d and h. The constant acceleraton formulas gve 0 V at V 1 d Vt at V ( L h) / mg( L d) a [ POINTS] 4. The fgure shows an expermental apparatus for measurng the resttuton coeffcent of, e.g. a golfball or a bowlng ball. It uses the followng procedure pendulum (a golf-club head, e.g.) s swung to a known ntal angle 1 and then dropped from rest so as to strke the ball The angle of follow-through of the pendulum s recorded Your goal s to derve a formula that can be used to determne the resttuton coeffcent e from the measured data. m 1 l v m 1 m m 4.1 Usng energy conservaton, derve an expresson for the speed V of the mass on the end of the pendulum ust before t strkes the ball, n terms of 1, l and the gravtatonal acceleraton. Energy conservaton gves 1 mgl(1 cos 1) mv V gl(1 cos 1) [ POINTS] 4. Smlarly, derve an expresson for the speed v 1 of the mass on the end of the pendulum ust after t strkes the ball, n terms of, l and the gravtatonal acceleraton. v gl(1 cos ) 1 [1 POINT] 4.3 Use momentum conservaton to calculate an expresson for the velocty v of the ball ust after t s struck, n terms of V and v 1, and any other necessary parameters. Momentum conservaton durng the collson requres that m V m v m v v m V v / m [ POINTS] 4

5 4.4 Hence, deduce a formula for the coeffcent of resttuton, n terms of 1,, l, and g, and any other necessary parameters. The resttuton coeffcent formula gves v v e(0 V ) e ( v v ) / V 1 1 e m V v / ( Vm ) v / V m1 ( m1 m (1 cos ) v1 m1 ( m1 m ) gl ) e m m V m m gl(1 cos ) 1 m1 ( m1 m ) m m (1 cos ) 1 (1 cos ) [ POINTS] 5. The fgure shows an dealzaton of a motor mounted on a flexble vbraton-solaton support. The motor turns the shaft at constant angular speed, so that t. The motor s free to move vertcally, but s prevented from movng horzontally or rotatng. The vbraton solator can be dealzed as a sprng wth stffness k and unstretched length L 0. The mass of the motor and the shaft may be neglected. The goal of ths problem s to derve an equaton of moton for the vertcal poston of the motor h(t). d motor m k,l 0 h 5.1 Wrte down an expresson for the poston vector of the mass m n terms of h, d, and r d cos h d sn 5. Hence, calculate a formula for the acceleraton of the mass m n terms of d, and and the tme dervatves of h. [1 POINT] d h a d cos d sn dt [ POINTS] 5

6 5.3 Draw a free body dagram for the solated porton of the system shown n the fgure (you can draw forces and moments drectly on the fgure). Idealze the motor and shaft as a massless frame. Note that the motor s prevented from movng horzontally and s also prevented from rotatng. N M m mg F s [3 POINTS] 5.4 Hence, show that the equaton of moton for h s gven by d h k k h L 0 g d snt dt m m Newton s law gves d h N ( Fs mg) md cos m d sn dt The sprng force law s Fs k( h L0 ). Take the component and re-arrange to get the answer stated. [ POINTS] 6