y = yo cos wt l φ m A pendulum has a mass m connected by a rigid rod of length to a pivot point. An external force causes the pivot point to oscillate vertically, with y yo cos wt. Gravity is pointing downward. = a) [10 pts] Write the Lagrangian for the pendulum, in terms of the angular coordinate o.
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- straight line is always acted upon by a force from the centre of the path and directed Ex. 2: A particle of mass 8 gm moving in a and maximum acceleration, if the amplitude is towards the centre. Find its maximum velocity sse magnitude is 128 times the displacement 5 cm.If you did the previous question right, you hopefully got an expression for yo. You may notice that you can simplify the differential equation a little bit: d'y k (y – yo) dt2 т The parameter yo now plays the roll of the "relaxed length". A better term may be "equilibrium value for y". But mathematically, it's identical to a relaxed length with the spring as the only force. We continue using this equation: y(t) = Y0 + A cos(wt + y) Now, solve for A (in cm) with these parameters. Again, if you need more information, enter -100000. The parameters are: •m = 200 grams • Yo = (equilibrium value) = 40 cm • k = (spring constant) = 0.03 N/cmneed asap please
- Two blocks and three springs are Configured below. All motion is honzental. when the blocks are aū jest spoings are unstretched: Kz k3 Suppse mi=2m mzzm, K,=4k okzz k, Ksz 2k. Find the charactesistics freguencies Foo Small oscillations.Consider a square sheet with sides: a. The sheet behaves like a compund pendulum and oscillates around an axis from one of the corners of the sheet where its Perpendicular to it. Find the period of the pandelum.A uniform rod of length 2.0 m, and 3.2 kg is suspended from a pivot a distance 0.2 m above its center of mass. The angular frequency in rad/s for small oscillations is approximately equal to: Take g=9.8m/s2. State your answer to the nearest 0.01 of rad/s.
- An object is attached to a coiled spring. The object is pulled down (negative direction from the rest position) 9 centimeters, and then released. Write an equation for the distance d of the object from its rest position, after t seconds if the amplitude is 9 centimeters and the period is 6 seconds. The equation for the distance d of the object from its rest position is (Type an exact answer, using z as needed. Use integers or fractions for any numbers in the equation.) (? Enter your answer in the answer box. Save for Later 3:18 PM O Type here to search O 11/15/2020 PgUp PgDn F12 DII PrtScn Home F9 End F10 F11 Ins F4 F5 F6 F7 F8 F1 F2 F3 2$ & ) %23 %3D 3. 4. 5 6 7 8 E R Y U | [ TDevelop a Lagrangian for the double pendulum. You may need to make some assumptions to simplify the problem. You may also need to introduce some new variables to make the problem work. Make sure that is explained.A pendulum consists of a mass m and a massless stick of length l. The pendulum support oscillates horizontally with a position given by x = A cos (wt), as shown below. Use 0 (the angle of the pendulum as measured from the vertical) as your generalized coordinate. In HW 8, you found the Lagrangian of this system is: 1 L = ·m (1² j² + x² + 2lxė cos 0 ) + mgl cos 0 m Use the Euler-Lagrange equation to find the equation of motion for this system. Then make a small angle approximation cos 0 ~ 1. The equation you find should be recognizable as the equation for a driven oscillator. What is the solution for 0 (t) in the small angle approximation?