need help As we discussed earlier in Problem Set 4.2, any object or quantity that is moving with a periodic sinusoidal oscillation is said to exhibit simple harmonic motion.This motion can be modeled by the trigonometric functiony = A sin (@t)%3Dory = A cos (wt)where A and w are constants. The frequency, given byf = 1/periodrepresents the number of cycles (or oscillations) that are completed per unit time. The unit used to describe frequency is the Hertz, where1 Hz = 1 cycle per second.A mass attached to a spring is pulled downward and released. The displacement of the mass from its equilibrium position after t seconds is given by the function= A cos(wt), where d is measured in centimeters (see the figure). The length of the spring when it is shortest is 13 centimeters, and 23 centimeters when it isdlongest. If the spring oscillates with a frequency of 0.4 Hertz, find d as a function of t.d =

A mass attached to a spring is pulled downward and released. The displacement of the mass from its equilibrium position after t seconds is given by the function d = A cos(ot), where d is measured in centimeters (see the figure). The length of the spring when it is shortest is 13 centimeters, and 23 centimeters when it is longest. If the spring oscillates with a frequency of 0.4 Hertz, find d as a function of t. d =

A mass attached to a spring is pulled downward and released. The displacement of the mass from its equilibrium position after t seconds is given by the function d = A cos(ot), where d is measured in centimeters (see the figure). The length of the spring when it is shortest is 13 centimeters, and 23 centimeters when it is longest. If the spring oscillates with a frequency of 0.4 Hertz, find d as a function of t. d =

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter12: Coupled Oscillations

Section: Chapter Questions

Problem 12.16P

See similar textbooks

Similar questions

The one-dimensional harmonic oscillator has the Lagrangian L = mx˙2 − kx2/2. Suppose you did not know the solution of the motion, but realized that the motion must be periodic and therefore could be described by a Fourier series of the form x(t) =∑j=0 aj cos jωt, (taking t = 0 at a turning point) where ω is the (unknown) angular frequency of the motion. This representation for x(t) defines many_parameter path for the system point in configuration space. Consider the action integral I for two points t1 and t2 separated by the period T = 2π/ω. Show that with this form for the system path, I is an extremum for nonvanishing x only if aj = 0, for j ≠ 1, and only if ω2 = k/m.
Prove that using x(t) = Asin (ωt + ϕ) will produce the same results for the period for the oscillations of a mass and a spring. Why do you think the cosine function was chosen?
Consider a simple harmonic oscillator consisting of a one kilogram mass m' on a spring with spring/forceconstant k and length L'. If the mass of the spring ms is 9% of the attached mass, and k = 66 N/m,and if we determined the attached body is displaced 3 cm and given a downward velocity of 0.4 m/s -calculate,→ the frequency ω of the motion, ,→ and the amplitude A of the motion
Two springs of force constants k1 and k2 are attached to a block of mass m and to fixed supports as shown in Fig. 2. The table surface is frictionless. (a) If the block is displaced from its equilibrium position and is then released, show that its motion will be simple harmonic with angular frequency w = √(k1 + k2)/m. (b) Suppose the system is now submersed in a liquid with damping coefficient b , what is the condition that the block will return to itsequilibrium position without oscillation?Sketch a graph to show the behavior of the system in this case.
The equation of motion for a damped harmonic oscillator is s(t) = Ae^(−kt) sin(ωt + δ),where A, k, ω, δ are constants. (This represents, for example, the position of springrelative to its rest position if it is restricted from freely oscillating as it normally would).(a) Find the velocity of the oscillator at any time t.(b) At what time(s) is the oscillator stopped?
Let the initial position and speed of an overdamped, nondriven oscillator be x0 and v0, respectively. (a) Show that the values of the amplitudes A1 and A2 in Equation 3.44 have the values A1=2x0+v021 and A2=1x0+v021 where 1 = 2 and 2 = + 2. (b) Show that when A1 = 0, the phase paths of Figure 3-11 must be along the dashed curve given by x=2x, otherwise the asymptotic paths are along the other dashed curve given by x=1x. Hint: Note that 2 1 and find the asymptotic paths when t .
Consider a bob attached to a vertical string. At t = 0, the bob is displaced to a new position where thestring makes −8° with the vertical and is then released from rest. If the angular displacement θ(t) iswritten as a cosine function, then the phase constant φ is:
Explain the term amplitude as it relates to the graph of asinusoidal function.
Let's consider a simple pendulum, consisting of a point mass m, fixed to the end of a massless rod of length l, whose other end is fixed so that the mass can swing freely in a vertical plane. The pendulum's position can be specified by its angle Φ from the equilbrium position. Prove that the pendulum's potential energy is U(ϕ)=mgl(1−cosϕ). Write down the total energy E as a function of Φ and ϕ˙. Show that by differentiating E with respect to t you can get the equation of motion for Φ. Solve for Φ(t). If you solve properly, you should find periodic motion. What is the period of the motion?
Note: Because the argument of the trigonometric functions in this problem will be unitless, your calculator must be in radian mode if you use it to evaluate any trigonometric functions. You will likely need to switch your calculator back into degree mode after this problem.A massless spring is hanging vertically. With no load on the spring, it has a length of 0.24 m. When a mass of 0.59 kg is hung on it, the equilibrium length is 0.98 m. At t=0, the mass (which is at the equilibrium point) is given a velocity of 4.84 m/s downward. At t=0.32s, what is the acceleration of the mass? (Positive for upward acceleration, negative for downward)
If a mass m is attached to a given spring,its period of oscillation is T. If two such springs are connected endto end, and the same mass m is attached, (a) is its period greaterthan, less than, or the same as with a single spring? (b) Verify youranswer to part (a) by calculating the new period, T′, in terms ofthe old period T.
Simple Harmonic Motion is defined as a periodic motion of a point along a straight line, such that its acceleration is always towards a fixed point in that line and is proportional to its distance from that point. In other words, its acceleration is proportional to the displacement but acts in an opposite direction. Prove algebraically that the motion of a particle defined by x=cos2t + 2sin2t is simple harmonic.