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9. (13 points) A block of mass m is attached to a top of a spring...
A block of mass m is attached to a top of a spring (spring constant k)....
A block of mass m is attached to a top of a spring (spring constant k). The bottom of the spring is attached to a car of mass M that is free to move on a horizontal track. The spring is rigid enough that it is only able to move up and down, not side to side. See below picture. Illlllll car a. Write the Lagrangian in terms of x, y, i, and y. b. Write the Hamiltonian in terms...
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...
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...
1) Consider a block of mass M connected through the massless rigid rod to the massless...
1) Consider a block of mass M connected through the massless rigid rod to the massless circular track of radius a on a frictionless horizontal table (see the Figure). A particle of mass m is constrained to move on the vertical circular track. The distance between the center of the circular track and the center of mass of the block of mass M is constant and equal to L. Assume that there is no friction between the track and the...
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...
A block of mass m = 2.4kg is attached to a single spring of spring constant...
A block of mass m = 2.4kg is attached to a single spring of spring constant k = 4.3??Nmand allowed to oscillate on a horizontal, frictionless surface while restricted to move in the x-direction. The equilibrium position of the block is ?=0?x=0m. At time ?=0?t=0s the mass is at position ?=−3.0?x=−3.0m and moving with x-component of velocity ??=4.5??vx=4.5ms. Where is the block at time ?=6.8?t=6.8s? Answer in meters.
1) A block of unknown mass is attached to a spring of spring constant 9.4 N/m...
1) A block of unknown mass is attached to a spring of spring constant 9.4 N/m and undergoes simple harmonic motion with an amplitude of 10.2 cm. When the mass is halfway between its equilibrium position and the endpoint, its speed is measured to be 30.7 cm/s. Calculate the mass of the block. Answer in units of kg. 2) Find the period of the motion. Answer in units of s. 3) Calculate the maximum acceleration of the block. Answer in...
A block of mass m = 2.1 kg is attached to a single spring of spring...
A block of mass m = 2.1 kg is attached to a single spring of spring constant k = 4.3 and allowed to oscillate on a horizontal, frictionless surface while restricted to move in the x-direction. The equilibrium position of the block is x = 0m. At time t = 0s the mass is at position x =-0.7m and moving with x-component of velocity vx-1.79. what is the x- component of velocity at time t = 5.3s? Answer in meters...
A block of mass m = 3.5 kg is attached to a spring with spring constant...
A block of mass m = 3.5 kg is attached to a spring with spring constant k = 520 N/m. It is initially at rest on an inclined plane that is at an angle of θ = 21° with respect to the horizontal, and the coefficient of kinetic friction between the block and the plane is μk = 0.16. In the initial position, where the spring is compressed by a distance of d = 0.14 m, the mass is at...
7) A block of mass m, resting on a horizontal frictionless surface, is attached to one...
7) A block of mass m, resting on a horizontal frictionless surface, is attached to one end of a spring; the other end is fixed to a wall. It takes 3.6 J of work to compress the spring by 0.13 m. If the spring is compressed, and the mass is released from rest, it experiences a maximum acceleration of 15 m/s2. Find the value of (a) the mass of the block. As the block passes through its equilibrium position, a...
A block of mass m is attached to a top of a spring (spring constant k). The bottom of the spring is attached to a car of mass M that is free to move on a horizontal track. The spring is rigid enough that it is only able to move up and down, not side to side. See below picture. Illlllll car a. Write the Lagrangian in terms of x, y, i, and y. b. Write the Hamiltonian in terms...
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...
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...
1) Consider a block of mass M connected through the massless rigid rod to the massless circular track of radius a on a frictionless horizontal table (see the Figure). A particle of mass m is constrained to move on the vertical circular track. The distance between the center of the circular track and the center of mass of the block of mass M is constant and equal to L. Assume that there is no friction between the track and the...
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...
A block of mass m = 2.1 kg is attached to a single spring of spring constant k = 4.3 and allowed to oscillate on a horizontal, frictionless surface while restricted to move in the x-direction. The equilibrium position of the block is x = 0m. At time t = 0s the mass is at position x =-0.7m and moving with x-component of velocity vx-1.79. what is the x- component of velocity at time t = 5.3s? Answer in meters...
7) A block of mass m, resting on a horizontal frictionless surface, is attached to one end of a spring; the other end is fixed to a wall. It takes 3.6 J of work to compress the spring by 0.13 m. If the spring is compressed, and the mass is released from rest, it experiences a maximum acceleration of 15 m/s2. Find the value of (a) the mass of the block. As the block passes through its equilibrium position, a...
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