Figure Q1(b) shows a mass of 5kg, in a water tank, is connected with a spring. The mass is pulled down as far as x with an initial Force of Fo 4N. The spring coefficient and the damping coefficient are known, k 10N/m and b = 15kg/s, respectively (b) (i) Derive the empirical equation of the homogeneous solution (ii) Compute the homogeneous solution if initial conditions are given, y(0) = 1 and y'(0) = -7. If the function of Force at particular time is given as F(t) = Fo sin wžt. Determine the particular solution. (iii)
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- Problem 2: A 200 g oscillator is oscillating at 2.0 Hz in a vacuum chamber. When air is admitted, the amplitude drops 60% in 50 s. How many oscillations will have been completed when the amplitude is 30% of its initial value?A spring with spring constant k = 35 N/m is attached to the ceiling. A cylinder with mass m = 1 kg and diameter d = 5 cm is attached to its lower end. The cylinder is first held, such that the spring 1 is neither stretched nor compressed, then a tank of water is placed underneath with the surface of the water just touching the bottom of the cylinder. When released, the cylinder will oscillate a few times but, damped by the water, quickly reaches an equilibrium position. When in equilibrium, what length of the cylinder is submerged?A spring with a mass of 2 kg has natural length 0.5 m. A force of 25.6 N is required to maintain it stretched to a length of 0.7 m. Suppose that the spring is immersed in a fluid with damping constant c=40. (a) Using Newton's Second and Hooke's Law, derive the 2nd order differential equation for the following mass-spring system. equilibrium position Figure 3 for Q3. (b) If the spring starts from equilibrium and is stretched to a length of 0.7 m and then released with initial velocity 0.6 m/s, find the position of the mass at any time t.
- Task 1 2cos4t. Find the period of (a) The position of a mass in a spring-mass system is given as x(1) oscillations and initial conditions. (b) Consider a vertical spring mass-damper-system with mass m=2 kg, damping coefficient c = 5 N.s/m and stiffness k = 10 N/m. An external force fl1) = sin3t N is applied to the mass. (i) Find the static equilibrium position as measured from the un-stretched length of the spring (ii) Write the equation of motion. (iii) Find the natural and damped frequencies. (iv) In steady state, find the time delay in seconds between the mass position and At). (v) Find the steady state maximum acceleration.A 10.0 kg object oscillates at the end of a vertical spring that has a spring constant of 1.90 x 104 N/m. The effect of air resistance is represented by the damping coefficient b = 3.00 N-s/m. (a) Calculate the frequency of the dampened oscillation. Hz (b) By what percentage does the amplitude of the oscillation decrease in each cycle? % (c) Find the time interval that elapses while the energy of the system drops to 3.50% 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.
- Diatomic molecules are typically found to have a frequency of oscillation in the range of 1012 Hz to 1014 Hz. Provide an order of magnitude estimate for the spring constant of the harmonic potential energy for such molecules.need help with dA mass-spring motion is governed by the ordinary differential equation d²x dx +b + k(t)x= F(t), dt² dt m where m is the mass, b is the damping constant, k is the spring constant, and F(t) is the external force. We consider the initial conditions x(0) = 1 and x/(0) = 0. Assume the following numerical values for this part of the project: m = 1, k = 1/5, b= 1/5, and F(t) = cos yt. 1 (a) Read section 4.10. Explain what is the resonance frequency, and then compute the resonance frequency for this mass-spring system. (b) The ODE45-solver can be used to obtain the solution curves in MATLAB. Use the script Project2_Q2.m to plot the solutions and estimate the amplitude A of the steady response for y = 0.2, 0.42, 0.6, and 0.8. (c) The script also provide you with the graph of A versus y. For what frequency 7 is the amplitude the greatest? Is it equal to that you obtained in (a)?
- A small block of mass M = 850 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 = 3.5 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 = 1.5 cm and released.Randomized Variables M = 850 gD = 1.5 cmk = 3.5 N/m a) Calculate a value for the magnitude of the maximum acceleration amax of the blocks in m/s2. b) Write an equation for the largest spring constant kmax for which the upper block does not slip. c) Calculate a value for the largest spring constant kmax for which the upper block does not slip, in N/m.A car with a mass m = 1000.0kg is moving on a horizontal surface with a speed v = 20.0m/s when it strikes a horizontal coiled spring and is brought to rest when the spring is compressed by a distance d = 3.0m. Calculate the spring stiffness constant k by... (a) select appropriate common equations and make appropriate substitutions that are specific to the problem, and algebraically manipulate the equations to end with an algebraic expression for the variable the problem is asking you to solve for. (b) Show the numeric substitution of given quantities and show the final numeric result. (c) Draw a free body diagram for the system and define the given quantities.Round to two decimal places if necessary. A spring is stretched 5 centimeters by a 15 N weight. The weight is then pulled down an additional 8 centimeters and released. Neglect damping. Find the function u(t) for the position of the spring at any time t. u(t) =