A particle of mass m constrained to move on the surface of a cylinder defined by x² + y = R° . The particle is subject to a force directed toward the origin and proportional to the distance of the particle from the origin: F=-kr. (a) Use the Hamiltonian method to find the equations of motion. (b) Draw the phase diagram for the particle. HA = Johr
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- 5. A mass on a spring is shown below. Let y(t) be the displacement of the spring as a function of time t. Position A is the equilibrium position corresponding to y = 0. As was done in class, we are taking the positive direction to be downward. The graph of the solution to an initial value problem for this spring is shown. Answer the following questions about this system. A m B A: y=0 C (a) The spring's starting position (at t = 0) could be at i. Position A (Equilibrium) ii. Position B (Above Equilibrium) iii. Position C (Below Equilibrium) iv. Not enough information to tell (b) The spring's starting velocity (at t = 0) is i. positive ii. zero iii. negative iv. Not enough information to tell (c) Immediately after being set in motion at t = 0) the spring1. m m= 200g, k= 4.44 N/m, v[0) = 0.1 m/s to the right, x(0) = 0.05m to the left of the equilibrium position a. Derive the particular equation of motion of the mass m b. Solve for the position of the mass at t= 0.1s, t= 0.5s and t = 1s c. Draw a graph of x(t) from t= Os to t=1.5%, clearly showing the amplitude value, correct frequency, and initial conditions.1. 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?
- 4. Consider a mass of 500 g placed at the end of a spring with stiffness constant 100 N/m and hanging at rest. The mass is displaced downward by 1.0 cm and released from rest. When the motion sets in, a time-varying force F(t) = cos³ 2t is applied to the spring-mass system. Solve the nonhomogeneous 2nd order differential equation representing the motion. That is, solve for x(t). [Note: Take the downward direction as the negative x-axis.]Look at the below system. Using either the conservation of energy method or Lagrange's method, solve for the governing equation of motion for the system. Put a box around your final answer. Also, put a box around your equations for the potential and kinetic energy of the system. Assume the system's springs are initially unstretched (i.e., assume that there is no gravity until t = 0 [s]). K₁ Î E K₂ XFor the following system, obtain the mathematical model (Hint: Use superposition principle and derive the mathematical model for mass ml and mass m2): X2 k2 ki k3 m2 m1 b2
- 1 An object of mass 125 kg is released from rest from a boat into the water and allowed to sink. While gravity is pulling the object down, a buoyancy force of times the weight of the object is pushing the object up (weight = mg). If we assume that water 40 resistance exerts a force on the object that is proportional to the velocity of the object, with proportionality constant 10 N-sec/m, find the equation of motion of the object. After how many seconds will the velocity of the object be 90 m/sec? Assume that the acceleration due to gravity is 9.81 m/ sec2. Find the equation of motion of the object. X(t) = %3D2- find the center of mass, the velocity of the center of mass, the momentum, and the kinetic energy of the following system: do f(xd) xaf] m₁ = 1 kg T₁=1+2j+3 k v₁ = 2î+3ĵ m₂ = 1 kg T₂ = 1-j+ k v₂ = 2) + 3k2. 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 (0Please answer quick you are in a bumper car with a total mass of 200.0kg your friend is also in a bumper car with a total mass of 200.0kg.You are moving to the right at 1.00m/s. Your friend is moving to the left at 0.500m/s. Your car hits a spot on your friends car that has no bumper. Your bumper acts as a spring with a spring constant of 2.00x10^4N/m. Determine the velocity of your car when the bumper has raced a maximum compression of 0.0700m.Assume friction is negligible.Problem 1 Show that if the following problem has a solution it is unique -k = f in (0,L)× (0,T), (1) u(0,t) = g,(t),"(L,t) = g,(t),0The particle in the box exercise 21 is in its ground state of nx=ny=nz=1. a) find the probability that the particle will be found in the volume defined by 0<x>L/4, 0<y>L/4, 0<z>L/4. b) Do the same for L/2 instead of L/4. ,SEE MORE QUESTIONSRecommended textbooks for youElements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill EducationControl Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEYElements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill EducationControl Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY