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Problem 2) Consider a simple pendulum consisting of a bob of mass m suspended by a...
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)...
A) Write the Lagrangian for a simple pendulum consisting of a point mass m suspended at...
A) Write the Lagrangian for a simple pendulum consisting of a point mass m suspended at the end of a massless string of length l. Derive the equation of motion from the Euler-Lagrange equation, and solve for the motion in the small angle approximation. B) Assume the massless string can stretch with a restoring force F = -k (r-r0), where r0 is the unstretched length. Write the new Lagrangian and find the equations of motion. C) Can you re-write the...
Consider a simple pendulum consisting of a massive bob suspended from a fixed point by a...
Consider a simple pendulum consisting of a massive bob suspended from a fixed point by a string. Let T denote the time (period of the pendulum) that it takes the bob to complete one cycle of oscillation. How does the period of the simple pendulum depend on the quantities that define the pendulum and the quantities that determine the motion? [You need to perform a dimensional analysis to solve this one. Start by assuming T = k Inmpgq, where k...
Consider a simple pendulum of length / and mass m placed in a rail-road cart that...
Consider a simple pendulum of length / and mass m placed in a rail-road cart that has constant acceleration a in the positive x-direction. (Hint: This means that suspension point of the pendulum moves with acceleration a, this needs to be accounted for when considering motion of the pendulum) a) (11 pts.) Find the Lagrangian function of this pendulum. b) (11 pts.) Obtain Lagrange's equations of motion for this pendulum. c) (11 pts.) Find the Hamiltonian function of this pendulum....
A simple pendulum has a rod of length LL with a bob of mass m=0.100 kg...
A simple pendulum has a rod of length LL with a bob of mass
m=0.100 kg at the end of the rod. The rod is supposed to be very
light and all the mass is imagined to be concentrated in the
bob.
A) What is the tension in the string when the pendulum swings by
an angle θ=5.00∘?
B) What is the magnitude of the restoring force on the bob when
θ=5.00∘?
C) If you drop a perpendicular from the...
A simple pendulum (mass M and length L) is suspended from a cart of mass m...
A simple pendulum (mass M and length L) is suspended from a cart of mass m that moves freely along a horizontal track shown at right. You might find it helpful to introduce the dimensionless parameters η-m/M and wo- /g/L. a What are the normal frequencies of small oscillations of the system (0 <1)? b Find and describe the corresponding normal modes of the system. c The cart/pendulum systern is held at rest in the configuration x-0 and θ K...
(10 points) As shown below. pendulum bob of mass M. The Suspended by a rod of...
(10 points) As shown below. pendulum bob of mass M. The Suspended by a rod of length Land of such that the pendulum bob will be own below, a bullet of mass and speed w passes completely through a of mass M. The bulleteres with a speed of w/3. The pendulum bob is by a stiff rod of length and ble mass. What is the minimum value at the pendulum bob will barely swing through a complete vertical cine?
The diagram shows a simple pendulum consisting of a mass M suspended by a thin siring....
The diagram shows a simple pendulum consisting of a mass M suspended by a thin siring. The mass swings back and forth between + and - theta_0. The mass M is 1.31kg and the length of the pendulum is 87.0cm. If theta_0 = 51.7degree, calculate the kinetic energy of the mass when theta = 31.7degree.
The Problem The single pendulum Consider the single pendulum shown below. There is a bob at...
The Problem The single pendulum Consider the single pendulum shown below. There is a bob at the end of the pendulum, of mass T32 For soscillations in the case of a frictionless pivot and a rod with negligible mass, the motion of the pendulv time can be described by the second-order linear ODE where 0 is the angle between the rod and the vertical, g is gravity, E is the length of the rod Q1 By substituting (2) into the...
Suppose you have a pendulum consisting of a small bob of mass m on a long...
Suppose you have a pendulum consisting of a small bob of mass m on a long (massless) string. The figure below shows the pendulum endpoints as well as its equilibrium position as it swings back and forth. Below that, acceleration and velocity vectors during the motion are shown for five points of the motion; the endpoints, the equilibrium position and somewhere between the equilibrium position and each endpoint Left side Middle Right side Motion (bob is the system) Cl Cl...
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)...
Consider a simple pendulum consisting of a massive bob suspended from a fixed point by a string. Let T denote the time (period of the pendulum) that it takes the bob to complete one cycle of oscillation. How does the period of the simple pendulum depend on the quantities that define the pendulum and the quantities that determine the motion? [You need to perform a dimensional analysis to solve this one. Start by assuming T = k Inmpgq, where k...
Consider a simple pendulum of length / and mass m placed in a rail-road cart that has constant acceleration a in the positive x-direction. (Hint: This means that suspension point of the pendulum moves with acceleration a, this needs to be accounted for when considering motion of the pendulum) a) (11 pts.) Find the Lagrangian function of this pendulum. b) (11 pts.) Obtain Lagrange's equations of motion for this pendulum. c) (11 pts.) Find the Hamiltonian function of this pendulum....
A simple pendulum has a rod of length LL with a bob of mass
m=0.100 kg at the end of the rod. The rod is supposed to be very
light and all the mass is imagined to be concentrated in the
bob.
A) What is the tension in the string when the pendulum swings by
an angle θ=5.00∘?
B) What is the magnitude of the restoring force on the bob when
θ=5.00∘?
C) If you drop a perpendicular from the...
A simple pendulum (mass M and length L) is suspended from a cart of mass m that moves freely along a horizontal track shown at right. You might find it helpful to introduce the dimensionless parameters η-m/M and wo- /g/L. a What are the normal frequencies of small oscillations of the system (0 <1)? b Find and describe the corresponding normal modes of the system. c The cart/pendulum systern is held at rest in the configuration x-0 and θ K...
(10 points) As shown below. pendulum bob of mass M. The Suspended by a rod of length Land of such that the pendulum bob will be own below, a bullet of mass and speed w passes completely through a of mass M. The bulleteres with a speed of w/3. The pendulum bob is by a stiff rod of length and ble mass. What is the minimum value at the pendulum bob will barely swing through a complete vertical cine?
The diagram shows a simple pendulum consisting of a mass M suspended by a thin siring. The mass swings back and forth between + and - theta_0. The mass M is 1.31kg and the length of the pendulum is 87.0cm. If theta_0 = 51.7degree, calculate the kinetic energy of the mass when theta = 31.7degree.
The Problem The single pendulum Consider the single pendulum shown below. There is a bob at the end of the pendulum, of mass T32 For soscillations in the case of a frictionless pivot and a rod with negligible mass, the motion of the pendulv time can be described by the second-order linear ODE where 0 is the angle between the rod and the vertical, g is gravity, E is the length of the rod Q1 By substituting (2) into the...
Suppose you have a pendulum consisting of a small bob of mass m on a long (massless) string. The figure below shows the pendulum endpoints as well as its equilibrium position as it swings back and forth. Below that, acceleration and velocity vectors during the motion are shown for five points of the motion; the endpoints, the equilibrium position and somewhere between the equilibrium position and each endpoint Left side Middle Right side Motion (bob is the system) Cl Cl...
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