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A mass m attached to the end of a massless rod of length L is free...
2) A particle of mass m, is attached to a massless rod of length L which...
2) A particle of mass m, is attached to a massless rod of length L which is pivoted at O and is free to rotate in the vertical plane as shown below. A bead of mass my is free to slide along the smooth rod under the action of a spring of stiffness k and unstretched length Lo. (a) Choose a complete and independent set of generalized coordinates. (b) Derive the governing equations of motion. m2
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)...
2. Consider a particle of mass M attached to a rigid massless rod of fixed length...
2. Consider a particle of mass M attached to a rigid massless rod of fixed length R whose other end is fixed at the origin. The rod is free to rotate about its fixed point. (a) Give an argument why the Hamiltonian for the system may be written as 21 21 with/-MR2 (b) If the particle carries charge q, and the rotor is placed in a constant magnetic field B, what is the modified Hamiltonian? (e) What is the energy...
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...
Aball of mass m is attached to a massless swing of length L. The swing is...
Aball of mass m is attached to a massless swing of length L. The swing is released from an angle θ from the vertical and, when it is in the vertical Ball on a Swing h above the floor. The ball then undergoes a projectile motion, Find an expression for the range R of the ball. Your expression should be in terms of the given position, it hits a stop which detaches Find an expression for the speed the ball...
A massless rod with length L is attached to two springs at its two masses (both...
A massless rod with length L is
attached to two springs at its two masses (both m) at its two ends.
The masses are connected to springs. The springs can move in the
horizontal and vertical directions as shown in the figure and they
both have a stiffness k. Note that gravity acts. Assume the springs
are un-stretched when the rod is vertical. Find the equation of
motion for the system using 1. Newton’s second law 2. Conservation
of energy....
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless ...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless spring, equilibrium length-Lo and spring constant- k). The other end of the spring is attached to the ceiling. Initially the spring is un-sketched but is making an angle θ° with the vertical, the mass is released from rest, see figure below. Let the instantaneous length of the spring be r. Let the acceleration due to gravity be g celing (a) (10...
4(12 points) A uniform rod of length L and mass M is attached at one end...
4(12 points) A uniform rod of length L and mass M is attached at one end to a frictionless pivot and is free to rotate about the pivot in the vertical plane as in Figure. The rod is released from rest in the horizontal position. (a)What are the initial angular acceleration of the rod and the initial translational acceleration of its right end (as shown in Fig.a)? (b)What is its angular speed when the rod reaches its lowest position (as...
A vertical axis is free to rotate. A massless, horizontal rod is attached to the axis...
A vertical axis is free to rotate. A massless, horizontal rod is attached to the axis as shown. There are two small objects attached to the rod. One of the objects has mass m1 and is fixed At the end of the rod. The other. With mass m2, starts right, at the point where the rod is attached to the axle. The axle is given an angular velocity w0. at n time defined to be t = 0 the object...
Three forces are acting on a massless rod of length 6 m as shown in the...
Three forces are acting on a massless rod of length 6 m as shown
in the figure below. The rod and the forces are in the xy plane. If
α=50°, β=30°, and γ = 25°, the torque on the rod about the z-axis
(through the origin of the coordinate system at point C ) is:
A). 35.6 k Nm . B). 46.0 k Nm . C). 71.3 k Nm . D). 56.3 k
Nm
125 N 10 N 30 N
2) A particle of mass m, is attached to a massless rod of length L which is pivoted at O and is free to rotate in the vertical plane as shown below. A bead of mass my is free to slide along the smooth rod under the action of a spring of stiffness k and unstretched length Lo. (a) Choose a complete and independent set of generalized coordinates. (b) Derive the governing equations of motion. m2
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)...
2. Consider a particle of mass M attached to a rigid massless rod of fixed length R whose other end is fixed at the origin. The rod is free to rotate about its fixed point. (a) Give an argument why the Hamiltonian for the system may be written as 21 21 with/-MR2 (b) If the particle carries charge q, and the rotor is placed in a constant magnetic field B, what is the modified Hamiltonian? (e) What is the energy...
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...
Aball of mass m is attached to a massless swing of length L. The swing is released from an angle θ from the vertical and, when it is in the vertical Ball on a Swing h above the floor. The ball then undergoes a projectile motion, Find an expression for the range R of the ball. Your expression should be in terms of the given position, it hits a stop which detaches Find an expression for the speed the ball...
A massless rod with length L is
attached to two springs at its two masses (both m) at its two ends.
The masses are connected to springs. The springs can move in the
horizontal and vertical directions as shown in the figure and they
both have a stiffness k. Note that gravity acts. Assume the springs
are un-stretched when the rod is vertical. Find the equation of
motion for the system using 1. Newton’s second law 2. Conservation
of energy....
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless spring, equilibrium length-Lo and spring constant- k). The other end of the spring is attached to the ceiling. Initially the spring is un-sketched but is making an angle θ° with the vertical, the mass is released from rest, see figure below. Let the instantaneous length of the spring be r. Let the acceleration due to gravity be g celing (a) (10...
4(12 points) A uniform rod of length L and mass M is attached at one end to a frictionless pivot and is free to rotate about the pivot in the vertical plane as in Figure. The rod is released from rest in the horizontal position. (a)What are the initial angular acceleration of the rod and the initial translational acceleration of its right end (as shown in Fig.a)? (b)What is its angular speed when the rod reaches its lowest position (as...
A vertical axis is free to rotate. A massless, horizontal rod is attached to the axis as shown. There are two small objects attached to the rod. One of the objects has mass m1 and is fixed At the end of the rod. The other. With mass m2, starts right, at the point where the rod is attached to the axle. The axle is given an angular velocity w0. at n time defined to be t = 0 the object...
Three forces are acting on a massless rod of length 6 m as shown
in the figure below. The rod and the forces are in the xy plane. If
α=50°, β=30°, and γ = 25°, the torque on the rod about the z-axis
(through the origin of the coordinate system at point C ) is:
A). 35.6 k Nm . B). 46.0 k Nm . C). 71.3 k Nm . D). 56.3 k
Nm
125 N 10 N 30 N
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