The system shown in Figure Q1 consists of two interconnected masses mi and mz. Both springs of stiffness coefficients k, and k3 have one on their ends attached to non-moving walls. The spring of coefficient kz and the damper of constant b are placed between the two masses. The positions of these masses are represented by x1 and xz respectively. This system has an external force u, acting on m, that is considered the input of the system. For the described system find the following: a) Equations of motion.
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- 8. A weight of 0.5kg stretches a spring by 0.49m. The spring-mass system is submerged in delicious melted butter with a damping coefficient of y=4. The spring is then lowered by an additional 1.0m and released with velocity 0. There is no external force. Find the function which gives the location of the weight at time t. Note: I have designed this to work out nicely. If it's not working out nicely then you probably got some butter in your calculations.At an outdoor market, a bunch of bananas is set into oscillatory motion with an amplitude of 20.0 cm on a spring with a force constant of 16.0 N/m. It is observed that the maximum speed of the bunch of bananas is 43.0 cm/s. What is the weight of the bananas in newtons? NThe system composed of two springs and a particle with mass ? shown in the figure can beconsidered as a system with a single equivalent spring. Suppose you move the particle and stretchboth springs (since the force constants are not the same, it cannot be assumed that both springs arestretch the same, you must find a relationship between how much each spring stretches and the displacement of the spring.particle). Determine an expression for the equivalent spring constant in terms of the constantsof each spring.
- A 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)?If we attach two blocks that have masses m1 and m2 to either end of a spring that has a force constant k and set them into oscillation by releasing them from rest with the spring stretched, show that the oscillation frequency is given by v = (k/m)^1/2, where m = m1m2 /(m1 + m2) is the reduced mass of the system.A stiff spring k = 666 N/m has be attached to the floor vertically. A mass of 6.66 kg is placed on top of the spring as shown below and it finds a new equilibrium point. If the block is pressed downward and released it oscillates. If the compression is too big, however, the block will lose contact with the spring at the maximum vertical extension. Draw a free body diagram and find that extension at which the block loses contact with the spring.
- A block with a mass m = 2.5 kg is pushed into an ideal spring whose spring constant is k = 4520 N/m. The spring is compressed x = 0.066 m and released. After losing contact with the spring, the block slides a distance of d = 2.25 m across the floor before coming to rest. a. Write an expression for the coefficient of kinetic friction between the block and the floor using the symbols given in the problem statement and g (the acceleration due to gravity). (Do not neglect the work done by friction while the block is still in contact with the spring.)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.A mass-spring-dashpot system is modeled by the differential equation: x" + 4x + 5x = f(t) (a) What is the type of oscillatory motion of the mass? Explain. (b) Find the transient solution of the system. (c) Find the steady state solution of the system for f(t) = 4 cos wt for w= 1. Write your solution as C cos (wt -a). (d) Solve the initial value problem for z(0) = 12, x'(0) = 0. (e) Draw the steady state solution and describe the oscillation.
- K Question 13 of 32 A spring hangs from the ceiling with an unstretched length of xo = 0.99 m. A m¡ = 9.1 kg block is hung from the %3D spring, causing the spring to stretch to a length X = 1.18 m. Find the length x2 of the spring when a m2 = 3.3 kg block is hung from the spring. For both cases, all vibrations of the spring are allowed to settle down before any measurements are made. m2 X2 = %3D 0.04 m m,A 0.62-kg block attached to a spring with force constant 154 N/m is free to move on a frictionless, horizontal surface as in the figure below. The block is released from rest after the spring is stretched 0.13 m. At that instant, find the force on the block (magnitude and direction). At that instant, find its acceleration (magnitude and direction).A horizontal spring attached to a wall has a force constant of 720 N/m. A block of mass 1.90 kg is attached to the spring and oscillates freely on a horizontal, frictionless surface as in the figure below. The initial goal of this problem is to find the velocity at the equilibrium point after the block is released. (a) What objects constitute the system, and through what forces do they interact? (b) What are the two points of interest? (c) Find the energy stored in the spring when the mass is stretched 6.40 cm from equilibrium and again when the mass passes through equilibrium after being released from rest. x = 6.40 _____ J x = 0 ______J (e) Substitute to obtain a numerical value. (f) What is the speed at the halfway point?