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Problem 3 Consider a bead, which is free to move around the frictionless wire hoop, which is spinning at a fixed ra...
A bead of mass m slides without friction along a rotating wire in the shape of...
A bead of mass m slides without friction along a rotating wire in the shape of a parabola with zar2, as shown below. The wire is rotating around the z-axis with constant angular velocity w z=ar2 (a) (0.5 point) Determine the Lagrangian for the system in terms of the coordinate r b) (1 point) Apply the Lagrange Equations to obtain the equation of motion. You (c) (0.5 points) Suppose that the bead is moving in a perfect circle of radius...
Problem 5 (15 points) A small bead can slide without friction on a circular hoop that...
Problem 5 (15 points) A small bead can slide without friction on a circular hoop that is a vertical plane and has a radius of 0.100 m. The hoop rotates at a constant rate of 4.00 rev/sec (recall 1 rev = 2π rad) about a vertical diameter as shown in the figure below (a) Find the angle β at which the bead is in vertical equilibrium. (It has a radial acceleration toward the axis.) (b) Is it possible for the...
Question 3 3. Consider a plane pendulum consisting of a mass m suspended by a massless...
Question 3
3. Consider a plane pendulum consisting of a mass m suspended by a massless string of length I. Suppose that that time t-0 the pendulum is put into motion and the length of the string is shortened at a constant rate ot-a (ie. L(t)= Lo-at). Use the angle of the pendulum φ as your generalized coordinate. (a) (2 points) Obtain the Lagrangian and Hamiltonian for this system (b) (0.5 points) Is H conserved? How can you tell? (c)...
Objectives for Lab 3 (Free Body Diagrams, Circular Motion). Prove that forces are vectors and understand...
Objectives for Lab 3 (Free Body Diagrams, Circular Motion). Prove that forces are vectors and understand the difference between externally applied forces and internal reaction forces. 11 Explore uniform circular motion. Conduct an experiment to understand centripetal force and centripetal acceleration. 21 Procedure for objective # 1. Suspend a block on a string in a vertical position. Let one member of a team pull the block to the right. a) Draw a free body diagram of the block. In which...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
A bead of mass m slides without friction along a rotating wire in the shape of a parabola with zar2, as shown below. The wire is rotating around the z-axis with constant angular velocity w z=ar2 (a) (0.5 point) Determine the Lagrangian for the system in terms of the coordinate r b) (1 point) Apply the Lagrange Equations to obtain the equation of motion. You (c) (0.5 points) Suppose that the bead is moving in a perfect circle of radius...
Problem 5 (15 points) A small bead can slide without friction on a circular hoop that is a vertical plane and has a radius of 0.100 m. The hoop rotates at a constant rate of 4.00 rev/sec (recall 1 rev = 2π rad) about a vertical diameter as shown in the figure below (a) Find the angle β at which the bead is in vertical equilibrium. (It has a radial acceleration toward the axis.) (b) Is it possible for the...
Question 3
3. Consider a plane pendulum consisting of a mass m suspended by a massless string of length I. Suppose that that time t-0 the pendulum is put into motion and the length of the string is shortened at a constant rate ot-a (ie. L(t)= Lo-at). Use the angle of the pendulum φ as your generalized coordinate. (a) (2 points) Obtain the Lagrangian and Hamiltonian for this system (b) (0.5 points) Is H conserved? How can you tell? (c)...
Objectives for Lab 3 (Free Body Diagrams, Circular Motion). Prove that forces are vectors and understand the difference between externally applied forces and internal reaction forces. 11 Explore uniform circular motion. Conduct an experiment to understand centripetal force and centripetal acceleration. 21 Procedure for objective # 1. Suspend a block on a string in a vertical position. Let one member of a team pull the block to the right. a) Draw a free body diagram of the block. In which...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
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