Problem Set 8

# Problem Set 8 - Problem Set VIII— Assign Due November 8...

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Unformatted text preview: Problem Set VIII— Assign November 11 2006 Due November 8, 2006. Fall 2006 Physics 200a R. Shankar . Problem 5.2.4 from Basic Training in I\-'Iatliematics Problem 5.3.2 from BTM. (Do just ﬁrst part. i.e, just ﬁrst pair of 2:1 and .22.) Problem 5.3.3 from BTM . Problem 5.3.6 from BTM ﬁhpatxDI—I . Consider a particle attached to a spring executing a motion :1: = A sin(wt + (i) with A 2 .32 m. At if = 0. it is at :1: 2 —.07m and a velocity —2m/s. The total energy is 5.6.]. Find (i) (i, (ii) f the frequency, (iii) it: and (iv) m. 01 6. A mass m. moving horizontally at velocity 'Uu on a frictionless table strikes a spring of force constant It. It compresses the spring and then bounces back with opposite velocity. Assuming no loss of energy anywhere ﬁnd out how long the mass is in contact with the spring and (ii) the maximum compression of the spring. 7. A steel beam of mass M and length L is suspended at its midpoint by a cable and executes torsional oscillations. If two masses or are now attached to either end of the beam and this reduces the frequency by 10%, what is m/M? 8. Imagine a solid disc, (say a penny,as in Fig (T?), of mass Ila", radius R, standing vertically on a table. A tiny mass m of negligible size is 110w glued to the rim at the lowest point. When disturbed, the penny rocks back and forth without slipping. Show that the period of the Simple Harmonic Motion is T : 2?? (33.13.. 27219 Hint: Find a, the restoring torque per unit angular displacement. When m —> 0 what happens to T. Explain in physical terms. \‘H. 1 _ FIG. 1. The disc has mass i'li’, radius 1? and is standing on a table where it rolls without. slipping. The tiny dot at the bottom has no size and a mass m. 9. I am driving my car on a parkway which has bumps every 30 m apart. At what 10. 11. 12. speed must I be driving to experience violent shaking if the suspension in my car has a resonant frequency of 0.5 Hz? For following problems the symbols are deﬁned the following equations (1‘11? our — — , = F ‘ t malt2 +bdt +162" geos(w) w0=\|ﬁ Show that a driven oscillator has its maximum amplitude of vibration at a frequency to Log — (tag/27112). At what frequency does the velocity have the greatest amplitude? For a damped oscillator (not driven by any external force) ﬁnd the time T* after which the amplitude of oscillations drops to half its value in terms of b and m. Consider a damped oscillator with L = 32, m = .5, b = 1 in MKS units. (i) Find the solution with mm) = 2,1;(0) = 0. I suggest using symbols till the every end. (ii) Now add on a driving force F0 cos cut with F0 = 10N and w = 2m. Find the solution with 32(0) 2 250(0) 2 0. ...
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Problem Set 8 - Problem Set VIII— Assign Due November 8...

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