Physics 3210What is the Lagrangian of a pendulum (mass m, length l)? Assume the potential energy is zero when q is zero.For which of these systems could you use Lagange’s equations of motion? 1. A double pendulum: a pendulum (mass m, length l) has a second pendulum (mass m, length l) connected to its bob. 2. A projectile moves in two dimensions with gravity and air resistance. 3. A bead slides without friction on a circular, rotating wire.What would be a good choice of generalized coordinates for the double pendulum? (Assume the pendula are constrained to move in the x-y plane.)What is the kinetic energy of a particle sliding on the parabola y=x2?What is the generalized force of a particle sliding on the parabola y=x2?A bead of mass m slides on a circular wire of radius R. The wire rotates about a vertical axis with angular velocity W. What is the kinetic energy of the bead?A bead of mass m slides on a circular wire of radius R. The wire rotates about a vertical axis with angular velocity W. When the bead is at angle q, how high is the bead above the lowest point of the wire?A bead of mass m slides on a circular wire of radius R. The wire rotates about a vertical axis with angular velocity W. The equation of motion of the bead is What are the equilibrium value(s) of q?A bead of mass m slides on a circular wire of radius R. The wire rotates about a vertical axis with angular velocity W. The equation of motion for small motions about the equilibrium is What is the oscillation frequency of the bead?Which of these constraints is holonomic? 1. A particle is constrained to slide on the inside of a sphere. 2. A disk rolls without slipping down an inclined plane (in one dimension). 3. A disk rolls without slipping on a table (in two dimensions). 4. A moving car is constrained to obey the speed limit.A particle of mass m slides on the outside of a cylinder of radius a. A good choice of generalized coordinates is (r, q). What is the constraint equation?A particle of mass m slides on the outside of a cylinder of radius a. What is the force of constraint (in the radial direction) when q=0?A particle of mass m slides on the outside of a cylinder of radius a. What happens to the (magnitude of the) force of constraint as q increases?A particle of mass m slides on the outside of a cylinder of radius a. What are the kinetic and potential energy of the particle?A particle of mass m slides on the outside of a cylinder of radius a. What condition must be satisfied by the force of constraint at the point (angle q0) where the particle leaves the cylinder?Physics 3210Week 2 clicker questionsWhat is the Lagrangian of a pendulum (mass m, length l)? Assume the potential energy is zero when  is zero. A.B.C.D.m( )( )2 21t m mg 1 cos2, ,Lq q = q - - q& &l l( )( )2 21t m mg 1 cos2, ,Lq q = q + - q& &l l( )( )21t m mg 1 cos2, ,Lq q = q + - q& &l( )( )21t m mg 1 cos2, ,Lq q = q - - q& &lFor which of these systems could you use Lagange’s equations of motion?1. A double pendulum: a pendulum (mass m, length l) has a second pendulum (mass m, length l) connected to its bob.2. A projectile moves in two dimensions with gravity and air resistance.3. A bead slides without friction on a circular, rotating wire.A. 1 onlyB. 2 onlyC. 3 onlyD. 1 and 2E. 1 and 3What would be a good choice of generalized coordinates for the double pendulum? (Assume the pendula are constrained to move in the x-y plane.)A. the Cartesian coordinates of the bob positions: x1, y1 (first bob) and x2, y2 (second bob)B. the Cartesian coordinates of the first bob: x1, y1 and the Cartesian coordinates of the second bob, treating the first bob as the origin: x’2, y’2C. the angles made between each pendulum rod and the vertical: 1, (first bob) 2 (second bob)D. the angles made between a line drawn from each pendulum bob to the pivot and the vertical: 1, (first bob) 2 (second bob)What is the kinetic energy of a particle sliding on the parabola y=x2?A.B.C.D.E. ( )21T x x t mx2, , =& &( ) ( )21T x x t mx 1 x2, , = +& &( ) ( )21T x x t mx 1 4x2, , = +& &( )( )2 21T x x t mx 1 2x2, , = +& &( )( )2 21T x x t mx 1 4x2, , = +& &What is the generalized force of a particle sliding on the parabola y=x2?A.B.C.D.E. 24mxx 2mgx-&2mgx-mgx-22mxx mgx-&24mx x 2mgx-&A bead of mass m slides on a circular wire of radius R. The wire rotates about a vertical axis with angular velocity . What is the kinetic energy of the bead?A.B.C.D. E. ( )2 2 2 21 1T t mR mR2 2, ,q q = q + W& &( )2 2 2 2 21 1T t mR mR cos2 2, ,q q = q + qW& &( )2 2 2 2 21 1T t mR mR sin2 2, ,q q = q + qW& &( )2 2 2 21 1T t mR mR2 2, ,q q = q - W& &( ) ( )221T t mR cos2, ,q q = q+W q& &A bead of mass m slides on a circular wire of radius R. The wire rotates about a vertical axis with angular velocity . When the bead is at angle, how high is the bead above the lowest point of the wire?A.B.C.D. E. h Rcos= qh Rsin= q( )h R sin cos= q- q( )h R 1 sin= - q( )h R 1 cos= - qA bead of mass m slides on a circular wire of radius R. The wire rotates about a vertical axis with angular velocity . The equation of motion of the bead isWhat are the equilibrium value(s) of ? A.B.C.D. E. 2gsin sin cos 0Rq+ q- W q q=&&12gcosR-� �q=� �W� �0q=12gsinR-� �q=� �W� �12g0 and cosR-� �q= q=� �W� �12g0 and sinR-� �q= q=� �W� �A bead of mass m slides on a circular wire of radius R. The wire rotates about a vertical axis with angular velocity . The equation of motion for small motions about the equilibrium isWhat is the oscillation frequency of the bead? A.B.C.D. E. 102gcosR-� �q =� �W� �2 20sin 0dq+W q dq=&&w=Wsinw=W q0sinw=W q2w=W2 20sinw=W qWhich of these constraints is holonomic?1. A particle is constrained to slide on the inside of a sphere.2. A disk rolls without slipping down an inclined plane (in one dimension).3. A disk rolls without slipping on a table (in two dimensions).4. A moving car is constrained to obey the speed limit. A. NoneB. Only oneC. Exactly twoD. Exactly threeE. All fourA particle of mass m slides on the outside of a cylinder of radius a. A good choice of generalized coordinates is (r, ). What is …