A point mass with mass m slides on a smooth rod. The rod rotates
with a constant 
around the origin on an x-y plane. Assume no external
forces. 
a) Find Lagrangian of the point mass and the equation of constraint.
b) Find Lagrange's equation of motion and eliminate the constraint.
c) Write the Lagrange's equation with undetermined multipliers and determine the constraint force.
Homework Answers
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A point mass with mass m slides on a smooth rod. The rod rotates
with a...
A bead of mass m slides frictionlessly on a circle of wire with radius R. The...
A bead of mass m slides frictionlessly on a circle of wire with radius R. The circle stands up in a vertical plane and rotates about the z-axis with constant angular velocity . Write down the Lagrangian. Find the equations of motion. For an angular velocity greater than some critical angular velocity , the bead will experience small oscillations about some stable equilibrium point . Find and (). We were unable to transcribe this imageWe were unable to transcribe this...
A rod of length L, mass m , and R resistance slides down with no friction....
A rod of length L, mass m , and R resistance slides down with no
friction. A set of two conducting, parallel railings with
negligible resistance (held at an angle to the ground). A constant
magnetic field B is present everywhere and is directed upwards
(perpendicular to the ground)
For the following values:
What is the velocity of the rod at equilibrium ? (g=10m/s^2)
We were unable to transcribe this imageWe were unable to transcribe this image
A rod of length L, mass m , and R resistance slides down with no friction....
A rod of length L, mass m , and R resistance slides down with no
friction. A set of two conducting, parallel railings with
negligible resistance (held at an angle to the ground). A constant
magnetic field B is present everywhere and is directed upwards
(perpendicular to the ground)
For the following values:
What is the velocity of the rod at equilibrium ? (g=10m/s^2)
We were unable to transcribe this imageWe were unable to transcribe this image
JUST ANSWER PART B A. A point mass m moves frictionlessly on a horizontal plane. An...
JUST ANSWER PART B
A. A point mass m moves frictionlessly on a horizontal plane. An unusual, anharmonic spring with unstretched length ro is attached between a pivot at the origin and the mass. Let the radial force exerted by the spring be given by Fr =-c(r-ro)" where c is a positive constant. Using plane polar coordinates r and θ: (i) Write down the Lagrangian L(r, θ,0) and use Lagrange's method to find the equations of motion for the mass...
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Find the Lagrangian. 2. (15 points) A bead of mass m slides under the influence of...
Find the Lagrangian.
2. (15 points) A bead of mass m slides under the influence of gravity along a straight wire. The wire can pivot around the support point at the bottom so that the angle a between the wire and the vertical can change. In addition, the wire rotates around vertical axis with a constant angular velocity w
A bead of mass m slides without friction along a rod, one end of which is...
A bead of mass m slides without friction along a rod, one end of which is pivoted in such a way that the rod can be revolved about the z-axis at a constant angle a, as shown in Fig. 2. The rod is driven with constant angular velocity w about Oz. Use Lagrange method to derive the equation of motion for the bead. Use the distance of m from the origin as generalized coordinate and discuss the motion of the...
Platform D rotates with an angular speed determined by motor M1, as shown in the figure....
Platform D rotates with an angular speed determined by motor M1, as shown in the figure. Attached to the platform is the drive M2 that rotates the smooth rod pa around the horizontal axis p in some particular way. Rod pa and verticals pb and ac are all on the same plane. A "bead" of mass m can slide freely along the rod, under the action of gravity. Assuming that α and θ are known functions of time, the system...
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Two cylinders, each having mass m, are linked by a connecting rod whose mass is negligible. A hor...
Two cylinders, each having mass m, are linked by a connecting rod whose mass is negligible. A horizontal force F(t) is applied to the right cylinder, and neither cylinder slips in its rolling motion. In the initial position, the angle θ locating the connecting pin is zero. Derive the equation of motion for this angle. We were unable to transcribe this image
Two cylinders, each having mass m, are linked by a connecting rod whose mass is negligible. A horizontal...
A rod of length L, mass m , and R resistance slides down with no
friction. A set of two conducting, parallel railings with
negligible resistance (held at an angle to the ground). A constant
magnetic field B is present everywhere and is directed upwards
(perpendicular to the ground)
For the following values:
What is the velocity of the rod at equilibrium ? (g=10m/s^2)
We were unable to transcribe this imageWe were unable to transcribe this image
A rod of length L, mass m , and R resistance slides down with no
friction. A set of two conducting, parallel railings with
negligible resistance (held at an angle to the ground). A constant
magnetic field B is present everywhere and is directed upwards
(perpendicular to the ground)
For the following values:
What is the velocity of the rod at equilibrium ? (g=10m/s^2)
We were unable to transcribe this imageWe were unable to transcribe this image
JUST ANSWER PART B
A. A point mass m moves frictionlessly on a horizontal plane. An unusual, anharmonic spring with unstretched length ro is attached between a pivot at the origin and the mass. Let the radial force exerted by the spring be given by Fr =-c(r-ro)" where c is a positive constant. Using plane polar coordinates r and θ: (i) Write down the Lagrangian L(r, θ,0) and use Lagrange's method to find the equations of motion for the mass...
Find the Lagrangian.
2. (15 points) A bead of mass m slides under the influence of gravity along a straight wire. The wire can pivot around the support point at the bottom so that the angle a between the wire and the vertical can change. In addition, the wire rotates around vertical axis with a constant angular velocity w
A bead of mass m slides without friction along a rod, one end of which is pivoted in such a way that the rod can be revolved about the z-axis at a constant angle a, as shown in Fig. 2. The rod is driven with constant angular velocity w about Oz. Use Lagrange method to derive the equation of motion for the bead. Use the distance of m from the origin as generalized coordinate and discuss the motion of the...
Platform D rotates with an angular speed determined by motor M1, as shown in the figure. Attached to the platform is the drive M2 that rotates the smooth rod pa around the horizontal axis p in some particular way. Rod pa and verticals pb and ac are all on the same plane. A "bead" of mass m can slide freely along the rod, under the action of gravity. Assuming that α and θ are known functions of time, the system...
Two cylinders, each having mass m, are linked by a connecting rod whose mass is negligible. A horizontal force F(t) is applied to the right cylinder, and neither cylinder slips in its rolling motion. In the initial position, the angle θ locating the connecting pin is zero. Derive the equation of motion for this angle. We were unable to transcribe this image
Two cylinders, each having mass m, are linked by a connecting rod whose mass is negligible. A horizontal...
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