The two-degree-of-freedom system as shown consists of a pendulum pm with mass m, and mass m2, connected together by two linear springs. Use the Hamilton's Principle to derive the equations of motion for the system. Assume small displacement and small rotation, i.e., 0 « 1, so that sin 0 z tan 0 × 0. k1 k2 L wm2 w (m1
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- One of the scientists, in trying to represent the relationship between the oil stiffness and the mechanical energy lost, writes down the equation 2 Elost = CS2, where C is a constant with appropriate units. Another scientist points out that this equation cannot be correct. Give two reasons why the equation cannot be correct.A pulley P is attached to the ceiling at O by a piece of metal that can swing freely. One end of a rope is attached to the ceiling at A. The rope is passed through the pulley P and a weight is attached to the other end of the rope at M, as shown in the diagram. A M The distance OA is 1 m, the length of the rope is 2 m, and the length of the piece of metal OP=r metres, where 0A frictionless pendulum of mass m=1 kg and length L=1 m, gets released at t=0 from a distance x=0.12 m relative to its equilibrium position. м M What is the velocity of the pendulum's mass at time t=1.5 s? Give your answer in units of [m/s] and up to 1 decimal place (e.g. 0.6)Determine the conditions under which mechanical energy is conserved in a pendulum. Consider a situation in which a pendulum made from a wooden sphere. The mass of the pendulum is 100 g and the length 1 m. If it is hanging on the ceiling of a room, it is made to oscillate. What, if anything, is the mechanical energy of the pendulum conserved? Make the argument grounded in physics.What happens to the frequency of oscillations of a physical pendulum when the amplitude of oscillations increases by a factor of 2? O Oscillation frequency increases by a factor of 4. O Oscillation frequency decreases by a factor of 2. O Oscillation frequency stays the same. O Oscillation frequency increases by a factor of 2. What happens to the energy of oscillations in this case? O Energy of oscillations increases by a factor of 2. O Energy of oscillations increases by a factor of 4. O Energy of oscillations stays the same. O Energy of oscillations decreases by a factor of 2.Show that the function x(t) = A cos ω1t oscillates with a frequency ν = ω1/2π. What is the frequency of oscillation of the square of this function, y(t) = [A cos ω1t]2? Show that y(t) can also be written as y(t) = B cos ω2t + C and find the constants B, C, and ω2 in terms of A and ω1H In a damped oscillator, let m = 250 g, k=85 N/m, and b=0.070 kg/s. In how many periods of oscillation the mechanical energy of the oscillator drop to one-half of its initial value?Two blocks of masses m1=1.0 kg and m2=3 kg are connected by an ideal spring of force constant k=4 N/m and relaxed length L. If we make them oscillate horizontally on a frictionless surface, releasing them from rest after stretching the spring, what will be the angular frequency ω of the oscillation? Choose the closest option. Hint: Find the differential equation for spring deformation.A point-like mass m is constrained to move along the z-axis only. It is attached to two springs, each of spring constant k, and each of rest length 0. One spring has one of its ends fixed at x = 0 and y = h; the other spring is fixed at one end to z = 0 and y=-h as shown in the figure below. Gravity does not act on the system. The springs do not bend but contract and extend along straight lines between their end points. y m (d) Determine the equilibrium position of the system and the frequencies of small oscillations about this equilibrium position. Hint: use approximations (i.e. Taylor expansion) to simplify the equation of motion near the equilibrium position to that of a harmonic oscillator. (e) Repeat part (d) for the case where the two springs have different spring constants k₁ and k₂ and are placed at different distances on the y-axis h₁ and h₂.Chapter 15, Problem 24 Z Your answer is partially correct. Try again. In the figure, two springs are joined and connected to a block of mass 40.3 kg that is set oscillating over a frictionless floor. The springs each have spring constant k = 242 N/m. What is the frequency (in Hz) of the oscillations? Number Units THz the tolerance is +/-2%A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring whose force constant is 83.8 N/m as shown. A horizontal force F→ causes the spring to stretch a distance of 5.46 cm from its equilibrium position. (a) Find the magnitude of F→ . (b) What is the total energy stored in the system when the spring is stretched? (c) Find the magnitude of the acceleration of the block just after the applied force is removed. (d) Find the speed of the block when it first reaches the equilibrium position. (e) If the surface is not frictionless but the block still reaches the equilibrium position, would your answer to part (d) be larger or smaller? (f) What other information would you need to know to find the actual answer to part (d) in this case? (g) What is the largest value of the coefficient of friction that would allow the block to reach the equilibrium position?An ideal spring is attached to the ceiling. While the spring is held at its relaxed length, a wooden block (M = 850 g) is attached to the bottom of the spring, and a ball of clay (m = 210 g) is pressed onto the bottom of the block so that they stick together. The block+clay are gently lowered through a distance of d = 2.7 cm and are then released, at which point they hang motionlessly from the bottom of the spring. After a few minutes have passed, the clay unsticks itself from the block and falls from rest to the ground. 1. What is the resulting period T of the block’s oscillation after the separation occurs? 2. As d approaches 0, what limit does T approach?SEE MORE QUESTIONS