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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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- Using 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_forwardA spring of unstretched length L and spring constant k is k attached to a wall and an object of mass M resting on a frictionless surface (see figure). The object is pulled such that M the spring is stretched a distance A then released. Let the +x- direction be to the right. L A 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 sin(wr + ) x (t) = make the phase angle as simple as possible. 2 Incorrect 4 T, where T is 3 What is the velocity v, of the object at t = V3 Aw 2 the period? Write the expression using fractions, not Ux = decimal values. Incorrectarrow_forwardNeed help on this problem. See attatched picture.arrow_forward
- A block of mass m = 0.32 kg is set against a spring with a spring constant of k1 = 681 N/m which has been compressed by a distance of 0.1 m. Some distance in front of it, along a frictionless surface, is another spring with a spring constant of k2 = 466 N/m. 1. Now assume friction is present on the surface in between the ends of the springs at their equilibrium lengths, and the coefficient of kinetic friction is μk = 0.5. If the distance between the springs is x = 1 m, how far d2, in meters, will the second spring now compress?arrow_forwardA 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_forward
- A 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_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_forwardA small sphere is attached to one end of a string of length 1.67 mm. The other end of the string is attached to the ceiling, and, with the string taut, the sphere is initially positioned so that the string attached to it makes an angle of 28.1º with the vertical. If the sphere is then let go (from rest) and allowed to swing on the string like a pendulum, what is the speed of the sphere at the instant it passes through the lowest point in its motion? (Neglect air resistance.)arrow_forward
- A heavy crate (m = 23 kg) is resting on a frictionless inclined plane while suspended from a wall by a spring.The plane is inclined at an angle of 30◦ from the horizontal, and the spring is stretched from its equilibriumposition by 25 cm. What is the spring constant of the spring? If the frictionless inclined plane is now replacedwith an otherwise identical plane except with a rough surface, the crate now stretches the spring by 12 cmfrom its equilibrium position. What is the coefficient of static friction? You can assume that the staticfriction is at its maximum value.arrow_forwardProve that when the average is taken with respect to position over one cycle, the average potential energy equals kA^2/6 and the average kinetic energy equals kA^2/3.arrow_forwardLet'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?arrow_forward
- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage LearningPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning