A thin hoop of radius R and mass M oscillates in its own plane with one point of the loop fixed. Attached to the hoop is a small mass M constrained to move (in a frictionless manner) along the hoop. Consider only small oscillations, and show that 2g V R the eigenfrequencies are @, = ,W, = 2R
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- A small block with mass m slides along a frictionless horizontal surface with a speed v = with and sticks to the end of a uniform rod (mass M = m, length L = 0.75 m) that can rotate around a frictionless axle through its upper end as shown. :0.4 m/s. It collides %3D a) Use Newton's 2nd law to find the angular frequency w of small angle oscillations for the combined system. b) What is the amplitude (maximum displacement 0max) from the vertical, expressed in degrees?Consider the solution tothe harmonic oscillator given above by x(t)=Ccos(wt−v) Prove tha tx(t0)=x(t0+2piw) In other words, the solution has the same value at time:t0 and at time:t0+2piw regardless of what value we have for ?0. The value 2piw is then the period T of the harmonic oscillator.Quartic oscillations Consider a point particle of mass m (e.g., marble whose radius is insignificant com- pared to any other length in the system) located at the equilibrium points of a curve whose shape is described by the quartic function: x4 y(x) = A ¹ Bx² + B² B²), (1) Where x represents the distance along the horizontal axis and y the height in the vertical direction. The direction of Earth's constant gravitational field in this system of coordinates is g = −gŷ, with ŷ a unit vector along the y direction. This is just a precise way to say with math that gravity points downwards and greater values of y point upwards. A, B > 0. (a) Find the local extrema of y(x). Which ones are minima and which ones are maxima? (b) Sketch the function y(x). (c) What are the units of A and B? Provide the answer either in terms of L(ength) or in SI units. (d) If we put the point particle at any of the stationary points found in (a) and we displace it by a small quantity³. Which stationary locations…
- A particle of mass m is suspended from a support by a light string of length which passes through a small hole below the support (see diagram below). The particle moves in a vertical plane with the string taut. The support moves vertically and its upward displacement (measured from the ring) is given by a function z = h(t). The effect of this motion is that the string-particle system behaves like a simple pendulum whose length varies in time. I b) [Expect to a few lines to wer these questions.] a) Write down the Lagrangian of the system. Derive the Euler-Lagrange equations. z=h(t) Compute the Hamiltonian. Is it conserved?Consider a hollow sphere (I = 2/3 M R2 when rotated about its center) of radius 0.49 m. The sphere is pinned at its north pole (this is not its center) at allowed to undergo small oscillations about this point. Calculate the period of the oscillation, is s, using g = 10 m/s2. (Please answer to the fourth decimal place - i.e 14.3225)Calculate the energy, corrected to first order, of a harmonic oscillator with potential: