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Walczyk
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I cannot solve these for the life of me. They are from my book, chapters regarding mechanics of rigid bodies, and lagrangian mechanics.. I need help badly, I'm having much trouble.
A small thin disk of radius r and mass m is attached rigidly to the face of a second thin disk of radius R and mass M as shown below. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle and released. Find the period of the motion.
A billard ball of radius a is initially spinning about a horizontal axis with angular speed w_0 and with zero forward speed. If the coefficient of slidign friction between the ball and the billiard table is µ_k, find the dsitance the ball travels before slipping ceases to occur.
The point of support of a simple pendulum is being elevated at a constant acceperation a, so that the height of the support is 1/2at^2, and its vertical velocity is at. Find the differential equation of motion for small oscillations of the pendulum by Lagrange's method. Show that the period of the pendulum is 2*pi*sqrt(l/(g+a)), where l is the length of the pendulum.
A heavy elastic spring of uniform stiffness and density supports a block of mass m. If m' is the mass of the spring and k its stiffness, show that the period of oscillations is 2*pi*sqrt((m+(m'/3))/k). This problem shows the effect of the mass of the spring on the period of oscillation. (Hilt: To set up the Lagrangian function for the system, assume that the velocity of any part of the spring is proportional to its distance from the point of suspension.)
A small thin disk of radius r and mass m is attached rigidly to the face of a second thin disk of radius R and mass M as shown below. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle and released. Find the period of the motion.
A billard ball of radius a is initially spinning about a horizontal axis with angular speed w_0 and with zero forward speed. If the coefficient of slidign friction between the ball and the billiard table is µ_k, find the dsitance the ball travels before slipping ceases to occur.
The point of support of a simple pendulum is being elevated at a constant acceperation a, so that the height of the support is 1/2at^2, and its vertical velocity is at. Find the differential equation of motion for small oscillations of the pendulum by Lagrange's method. Show that the period of the pendulum is 2*pi*sqrt(l/(g+a)), where l is the length of the pendulum.
A heavy elastic spring of uniform stiffness and density supports a block of mass m. If m' is the mass of the spring and k its stiffness, show that the period of oscillations is 2*pi*sqrt((m+(m'/3))/k). This problem shows the effect of the mass of the spring on the period of oscillation. (Hilt: To set up the Lagrangian function for the system, assume that the velocity of any part of the spring is proportional to its distance from the point of suspension.)