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The system with 2 degrees of freedom = x,y- generalised coordinates.
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Prob. 7.3: A simple pendulum (mass M and length L) is suspended from a cart (mass...
A plane pendulum of length L and mass m is suspended from a block of mass...
A plane pendulum of length L and mass m is suspended from a
block of mass M. The block moves without friction and is
constrained to move horizontally only (i.e. along the x axis). You
may assume all motion is confined to the xy plane. At t = 0, both
masses are at rest, the block is at
, and the pendulum has angular deflection
with respect to the y axis.
a) Using
and
as generalized coordinates, find the Lagrangian...
(25 points) anchored to two facing walls as shown in the figure. Inside the cart, a pendulum of mass m (not included in the mass M of the cart) and length l is hung from the ceiling, z is the dis...
(25 points) anchored to two facing walls as shown in the figure. Inside the cart, a pendulum of mass m (not included in the mass M of the cart) and length l is hung from the ceiling, z is the displacement of the cart from its equilibrium position, and ф is the angle the pendulum makes with the vertical. 4. Coupled oscillators. A cart of mass M when empty is attached to two springs (a) Write down the kinetic and...
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...
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 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 with mass m = 2.1 kg and length L = 2.3 m hangs...
A simple pendulum with mass m = 2.1 kg and length L = 2.3 m hangs from the ceiling. It is pulled back to an small angle of θ = 11.9° from the vertical and released at t = 0. 4)What is the angular displacement at t = 3.56 s? (give the answer as a negative angle if the angle is to the left of the vertical) 6)What is the magnitude of the radial acceleration as the pendulum passes through...
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...
A pendulum consists of a mass m suspended from a string of length L. When the...
A pendulum consists of a mass m suspended from a string of length L. When the pendulum is swinging, how much greater is the mass’s potential energy when the string is displaced from the vertical by the angle θ than when the mass is at the lowest position?
A simple pendulum with mass m = 2.3 kg and length L = 2.62 m hangs...
A simple pendulum with mass m = 2.3 kg and length L = 2.62 m
hangs from the ceiling. It is pulled back to an small angle of θ =
9.2° from the vertical and released at t = 0. 1) What is the period
of oscillation?
2) What is the magnitude of the force on the pendulum bob
perpendicular to the string at t=0?
3) What is the maximum speed of the pendulum?
4) What is the angular displacement...
A simple pendulum with mass m = 2.1 kg and length L = 2.79 m hangs...
A simple pendulum with mass m = 2.1 kg and length L = 2.79 m hangs from the ceiling. It is pulled back to a small angle of θ = 11.5° from the vertical and released at t = 0. 1) What is the period of oscillation? 2) What is the magnitude of the force on the pendulum bob perpendicular to the string at t=0? 3) What is the maximum speed of the pendulum? 4) What is the angular displacement...
A plane pendulum of length L and mass m is suspended from a
block of mass M. The block moves without friction and is
constrained to move horizontally only (i.e. along the x axis). You
may assume all motion is confined to the xy plane. At t = 0, both
masses are at rest, the block is at
, and the pendulum has angular deflection
with respect to the y axis.
a) Using
and
as generalized coordinates, find the Lagrangian...
(25 points) anchored to two facing walls as shown in the figure. Inside the cart, a pendulum of mass m (not included in the mass M of the cart) and length l is hung from the ceiling, z is the displacement of the cart from its equilibrium position, and ф is the angle the pendulum makes with the vertical. 4. Coupled oscillators. A cart of mass M when empty is attached to two springs (a) Write down the kinetic and...
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...
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....
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...
A simple pendulum with mass m = 2.3 kg and length L = 2.62 m
hangs from the ceiling. It is pulled back to an small angle of θ =
9.2° from the vertical and released at t = 0. 1) What is the period
of oscillation?
2) What is the magnitude of the force on the pendulum bob
perpendicular to the string at t=0?
3) What is the maximum speed of the pendulum?
4) What is the angular displacement...
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