21. A mass weighing 12 pounds stretches a spring 3 inches. What is the spring constant for this spring?
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According to the given data
W=12lb
x=3in
We need to find
Stiffness constant
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- 3. A vehicle towing a trailer through a spring-damper coupling hitch shown below. The following parameters and variables are defined: M is the mass of the trailer; K, the spring constant of the hitch, B, the viscous-damping coefficient of hitch; B, is the viscous-friction coefficient of the trailer; y,(t), the displacement of the towing vehicle; y.(t), the displacement of the trailer, and f(t), the force of the vehicle. TRAILER M B, a) Draw the free body diagram of the system, showing all the forces acting on the body. b) Write the differential equation of the system. c) Find the transfer functions9: A mass attached to a spring, set in a container of molasses (a thick liquid) acts like a damped simple harmonic oscillator. The mass is 16 kg, the spring has a spring constant of 64 N/m and the molasses provides a damping force of 64 N.s/m. (a) The system is (A) underdamped (B) overdamped (C) critically damped (b) A force of F(t) = -2e-2t (in N) is applied to the mass. If the method of undetermined coefficients is used to find a particular solution, which of the following is the best choice as a trial solution? (c) Initially, at t = 0s, our mass is moving at a speed of 3 m/s as it passes through its equilibrium position (x = 0m). If we are still under the force from part (b), at what positive time, in seconds, does it again pass through x = 0m? (If it never does, type dne).2. A kilogram mass is attached to the end of the spring with spring constant 2 N/m. Find the equation of motion if the mass is initially released (set in motion) from rest from a point 1 meter above equilibrium position. (Use the convention that displacements measured below the equilibrium position are positive.) (a) Write the initial-value problem which describes the position of the mass. (b) Find the solution to your initial-value problem from part (a). (c) Graph the solution found in (b) on (0Suppose a spring with spring constant 7 N/m is horizontal and has one end attached to a wall and the other end attached to a 2 kg mass. Suppose that the friction of the mass with the floor (i.e., the damping constant) is 1 N⋅s/m a) Set up a differential equation that describes this system. Let x to denote the displacement, in meters, of the mass from its equilibrium position, and give your answer in terms of x,x′,x′′. Assume that positive displacement means the mass is farther from the wall than when the system is at equilibrium. b) Find the general solution to your differential equation from the previous part. Use c1 and c2 to denote arbitrary constants. Use t for independent variable to represent the time elapsed in seconds. Your answer should be an equation of the form x=… c) Enter a value for the damping constant that would make the system critically damped. ?Ns/m1. In the laboratory, when you hanged 100 grams at the end of the spring it stretched 10 cm. You pulled the 100-gram mass 6 cm from its equilibrium position and let it go at t = 0. Find an equation for the position of the mass as a function of time t. 2. The scale of a spring balance found in an old Physics lab reads from 0 to 15.0 kg is 12.0 cm long. To know its other specifications, a package was suspended from it and it was found to oscillate vertically with a frequency of 2.00 Hz. Calculate the spring constant of the balance? (b) How much does the package weigh?1. Suppose you are riding your bicycle on a bumpy road having a surface profile that varies harmonically with +/- 6 cm undulations. The distance between consecutive peaks of these undulations is 2 m. When you sit on the seat of your bike for your casual ride, the springs deflect 5 cm. When you are seated, the damper under the seat provides an equivalent linear viscous damping of 10% of the critical damping. A simple representation of your ride on the "never-ending" rough road is shown below. (a) If you are riding your bicycle at a horizontal speed of 2.5 m/sec, how much bumping up and down will you experience? (b) Next day, you are carrying a backpack which increases your on-seat weight by 20%. Assuming that you are still able to ride with same speed, will this "loaded" ride be more or less comfortable than your previous, "no backpack" ride? M k/23 k/2 6 cm 2 m1. A 3 kg object is attached to spring and will stretch the spring 392 mm by itself. There is no damping in the system and a forcing function of the form F(t) = 10 cos (t) is attached to the object. If the object is initially displaced 20 cm downward from its equilibrium position and given a velocity of 10 cm/sec upward find the displacement at any time t. Show graph.62. •A 5-kg object is constrained to move along a straight line. Its initial speed is 12 m/s in one direction, and its final speed is 8 m/s in the opposite Complete the graph of force versus time with direction. F (N) (s) appropriate values for both variables (Figure 7-26). Several answers are correct, just be sure that your answer is internally consistent. Figure 7-26 Problem 621. Suppose we have a spring with attached mass hanging from the ceiling. The vertical displacement of the mass from its equilibrium (where it sits while at rest) can be modeled by the equation y(t) = e One can determine the coefficients a, b, c and 0, by solving a differential equation which incorporates the mass, spring stiffness, friction and so on, or, we can determine these coefficients by studying some data given by its motion. 1.5 1 0.5 -0.5 -1 Ty(t) e-at (b cos(t) + csin(t)). TA B с. The black curve is y(t), the position of the spring with respect to its equilibrium position. (a) Using the data A = (0, 1), B = (3,0), C = (2,-), D = (1, 0) and E = (1, 1), find the corresponding equation for y(t).A spring has an unstretched length of 12 cm. When an 80 g ball is hung from it, the length increases by 4.0 cm. Then the ball is pulled down another 4.0 cm and released.a. What is the spring constant of the spring?b. What is the period of the oscillation?c. Draw a position-versus-time graph showing the motion of the ball for three cycles of the oscillation. Let the equilibrium position of the ball be y = 0. Be sure to include appropriate units on the axes so that the period and the amplitude of the motion can be determined from your graph.An object attached to the end of a vertical helical spring bounces with a frequency of 2.1 Hz. If the spring constant is 5.9 N/m, what is the mass of the object?A fender is mounted on a automobile though dampers (to absorb collision energy) and springs (so that the fender can recover after low-speed collisions). During a crash-test, the automobile is moving at 2 m/s when its fender strikes a concrete barrier. The vehicle mass, m, is 1,000 kg. (In comparison, the fender itself is essentially massless.) The springs that mount the fender have a stiffness, k, of 1,000,000 N/m. y m barrier 1) Write a differential equation for the deflection of the springs when the fender is in contact with the barrier. 2) If the damping coefficient, c, is 30,000 N-s/m, what is the damping ratio of the mass-spring-damper system when the fender is in contact with the barrier? 3) For that damping coefficient, make a reasonably accurate sketch (with properly labeled axes) of the time-course of the force exerted on the barrier, starting from the moment of first contact. 4) Is there any value of the damping coefficient, c, that would yield no rebound of the vehicle from…SEE MORE QUESTIONS