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Expanding the integrand in Eq. (19.19) of Sec. 19.1C into a series of even powers of sin φ and integrating, show that the period of a simple pendulum of length l may be approximated by the formula where 0m is the amplitude of the oscillations.Reference to Equation 19.19:
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- A meter stick of total length l is pivoted a distance d from one end on a friction-less bearing. The stick is suspended so that it becomes a pendulum. This is called a "physical" pendulum because the mass is distributed over the body of the stick. Assume the total mass is m and the mass density of the stick is uniform. The acceleration of gravity is g. Find T and V as functions of generalized coordinate θ and velocity ˙θ. Do this by considering the stick to be divided into infinitesimal parts of length dl and integrating to find the total kinetic and potential energy. Set up the Lagrangian and find the equation of motion.Problem 11: A small block of mass M= 350 g is placed on top of a larger block of mass 3M which is placed on a level frictionless surface and is attached to a horizontal spring of spring constant k = 1.9 N/m. The coefficient of static friction between the blocks is μ =0.2. The lower block is pulled until the attached spring is stretched a distance D = 2.5 cm and released. Randomized Variables M = 350 g D = 2.5 cm k = 1.9 N/m Part (a) Assuming the blocks are stuck together, what is the maximum magnitude of acceleration amax of the blocks in terms of the variables in the problem statement? amax = k D/(3 M+M ) ✓ Correct! Part (b) Calculate a value for the magnitude of the maximum acceleration amax of the blocks in m/s². ✓ Correct! | @mar= 0.03390 Part (c) Write an equation for the largest spring constant kmax for which the upper block does not slip. Kmax = μ (M +M) g/klUse the following transformation to solve the linear harmonic oscillator problem: Q = p + iaq, P = (p − iaq) / (2ia)
- We can model a molecular bond as a spring between two atoms that vibrate with simple harmonic motion.The figure below shows an simple harmonic motion approximation for the potential energy of an HCl molecule.This is a good approximation when E < 4 ×10^−19. Since mH << mCl, we assume that the hydrogen atomoscillates back and forth while the chlorine atom remains at rest. Estimate the oscillation frequency of theHCl molecule using information in the figure below.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)(5) The periodic time of the physical pendulum is (4n*k/gl), where k is the radius of gyration.