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When an object is displaced by an amount x from stable equilibrium, a restoring force acts on it, tending to return the object to its equilibrium position. The magnitude of the restoring force can be a complicated function of x. In such cases, we can generally imagine the force function F(x) to be expressed as a power series in x as
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- In free space, E (z, t) = 1.0 sin (ωt-βz)ax ,(v/m). Show that the average power passing through a circular disk with a radius of 15.5 m in the constant plane z= is 1W.arrow_forwardThe mass m is attached to a spring of free length b and stiffness k. The coefficient of friction between the mass and the horizontal rod is u. The acceleration of the mass can be shown to be x=-f(x), where k f(x) = ug+(b + x) 1–- m b² +x² If the mass is released from rest at x = b, its speed at x = 0 is given by 1. Compute vo by numerical Simpson's 1/3 and 3/8 integration and compare between them with different step size, using the data m = 0.9 kg. b = 0.6 m, =0.3, k = 100 N/m, and g = 9.81 m/s. 2. Develop a MATLAB code to solve the equation for both methods. 3. Plot the acceleration of the mass versus x, and find the area under the curve by MATLAB built-in function. 4. Can we find an exact solution??Try it. www Figure (1)arrow_forwardUsing Gram-Schmidt algorithm, check for linear independence of the following vectors: v= (1, -2, 1, -1), v2= (1, 1, 3, -1), and v3= (-3, 7, 1, 3).arrow_forward
- A mass of 2 kg is attached to a spring. A force of 150 N is required to hold the spring stretched by 60 cm. The mass is in a medium that exerts a viscous resistance of 16 N when the mass has a velocity of 4 m/sec. Suppose the object is displaced (stretched) by 20 cm from equilibrium and released with no initial velocity. Find an equation for the object's position, u(t), in meters after t seconds.arrow_forwardA 54.0-kg box is being pushed a distance of 6.80 m across the floor by a force P→ whose magnitude is 197 N. The force P→ is parallel to the displacement of the box. The coefficient of kinetic friction is 0.178. Determine the work done on the box by each of the four forces that act on the box. Be sure to include the proper plus or minus sign for the work done by each force. (a) WP = (b) Wf = (c) Wmg = (d) WN =arrow_forwardA 7-lb collar is attached to a spring and slides without friction along a rod in the vertical plane. The spring has a constant 3.5 lb/in and 15-in. undeformed length. The dimensions are: h = 15 in. The collar is moving to the left at v = 2 in/s in the position shown. Determine the force exerted by the rod on the collar at (a) point A, (b) point B. Both A and B are on the curved portion of the rod. h LA h B h harrow_forward
- of a copper wire of uniform cross section and Ex. 70: find the energy stored per unit volume of length 1.5 m, when it is stretched to a length of a copper wire of uniform cross section and of length 1.5 m, when it is stretched to length a of 1.51 m by a stress of 3 x 102 N/m2.arrow_forwardA scallop forces open its shell with an elastic material called abductin, whose Young's modulus is about 2.0 x 106 N/m².arrow_forwardA spring of unstretched length L and spring constant k is attached to a wall and an object of mass M resting on a k frictionless surface (see figure). The object is pulled such that the spring is stretched a distance A then released. Let the +x-direction be to the right. M L. What is the position function x (t) for the object? Take x = 0 to be the position of the object when the spring is relaxed, and make the phase angle as simple as possible. x(1) = A sin) Incorrect What is the velocity v, of the object at t = T, where T is the period? Write the expression using fractions, not decimal values. Incerrect What is the acceleration a, of the object at r =arrow_forward
- A crate of mass m1�1 slides down a well-lubricated hill of height hℎ, with negligible friction. At the bottom, where it is moving horizontally, it collides with another crate, of mass m2, that initially was sitting at rest and that is attached to a wall by a spring of spring constant k that initially is at its equilibrium length. Assume that the spring itself has negligible mass. a) Write an equation that expresses the speed v of the crate when it arrives at the bottom of the hill in terms of the height of the hill and other known quantities. b) When the two crates collide, they stick together, due to Velcro pads pasted on their mating surfaces. Write an equation that expresses the speed v2 of the two crates after the collision, in terms of the speed v of crate m1 just before the collision. c)As the coupled crates continue to move to the right, they compress the spring by a distance d, at which point they momentarily stop. Assuming that the coefficient of kinetic friction between the…arrow_forwardA 0.4 Gram mass is required to stretch a soap film confined in arectangular frame by 2mm downward before the film breaks. If the length of the movable bar is 2.5 cm, what will be the surface tension and change in surface free energy of the soap film?arrow_forwardWe will now determine how the 1/3 rule comes about. Consider a spring of mass ms which is attached to a wall and oscillates on a frictionless surface as shown below. The spring’s mass is uniformly distributed along the length of the spring. We will start with the infinitesimal form of kinetic energy, i.e. dKE = ½ (dms )v2. This formula will apply to an infinitesimal segment of the spring of length dx and mass dms as indicated below. For any point on the spring, the velocity of oscillation will be given by v = (ve/L)x where ve is the velocity of the spring at its end where the mass m is attached, and L is the stretched length of the spring at that instant. Thus, when x = 0 then v = 0, and when x = L/2 then v = ½ ve. Hint: Figure out how to relate dms to dx and then integrate both sides of the infinitesimal kinetic energy equation to get an equation for the kinetic energy of the spring that includes ms/3.arrow_forward
- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning