A disk with mass Mand radius R is on a horizontal table. The disk can rotate...

 

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A disk with mass Mand radius R is on a horizontal table. The disk can rotate about a vertical axis through its centre, O. Figure 1 shows a top view of the disk. A point mass m is attached to the center O through a spring with spring constant *. The spring is in equilibrium when is a distance % from the origin O. M R Figure 1. A horizontal disk with mass Mand radius R can rotate without friction about an axis through its centre, perpendicular to the plane of the drawing. The force of gravity is into the plane of the drawing. The mass m moves without friction, radially on a massless track, subject to a force from the spring only. At a distance 70 from the origin O, the spring force is zero. This problem has two degrees of freedom. We choose generalized variables and, attached to the table top. We describe the system in an inertial frame. a) Write the Lagrangian of the system in terms of the specified generalized variables. b) Derive the Lagrangian equations of motion from the Lagrangian. c) Solve the Langrangian equations, to obtain an expression for the angular velocity of the disk as a function of time. Make use of a differential equation solved in PHYS 341, so you do not have to do (or show) that work! d) Make a graph of the angular velocity of the disk with the point mass as a function of time, for 0 A disk with mass Mand radius R is on a horizontal table. The disk can rotate about a vertical axis through its centre, O. Figure 1 shows a top view of the disk. A point mass m is attached to the center O through a spring with spring constant *. The spring is in equilibrium when is a distance % from the origin O. M R Figure 1. A horizontal disk with mass Mand radius R can rotate without friction about an axis through its centre, perpendicular to the plane of the drawing. The force of gravity is into the plane of the drawing. The mass m moves without friction, radially on a massless track, subject to a force from the spring only. At a distance 70 from the origin O, the spring force is zero. This problem has two degrees of freedom. We choose generalized variables and, attached to the table top. We describe the system in an inertial frame. a) Write the Lagrangian of the system in terms of the specified generalized variables. b) Derive the Lagrangian equations of motion from the Lagrangian. c) Solve the Langrangian equations, to obtain an expression for the angular velocity of the disk as a function of time. Make use of a differential equation solved in PHYS 341, so you do not have to do (or show) that work! d) Make a graph of the angular velocity of the disk with the point mass as a function of time, for 0