Lagrangian function and equation of motion
A bead slides without friction on a frictionless wire in the shape of a cycloid with equations
x = a(θ-sin θ)y = a(1+cos θ)
where θ's range is 0 to 2π.Find the Lagrangian function and the equation of motion.
Homework Answers
Lagrangian function
A bead slides without friction on a frictionless wire in the shape of a cycloid with equations x = a(θ-sin θ)y = a(1+cos θ) where θ's range is 0 to 2π.Find the Lagrangian function and the equation of motion.
A frictionless wire is bent into the shape of a cycloid curve, with coordinates given by...
A frictionless wire is bent into the shape of a cycloid curve,
with coordinates given by the parametric equations
? = ?(? + sin ?), ? = ?(1 − cos ?), for −? < ? < ?. The x
axis is horizontal, and y is vertically upwards. A bead of
mass m slides freely on the wire. Show that the distance s,
measured along the wire from the origin, is given by
? = 4? sin. Write out the potential...
A bead of mass m slides along a frictionless wire under the influence of gravity. The...
A bead of mass m slides along a frictionless wire under the influence of gravity. The shape of the wire is given by the equation y = axa, where x denotes the horizontal co-ordinate, y denotes the vertical co-ordinate, and a is a constant. (a) Use Lagrange's equation to determine the equation of motion. (b) Compute Hamilton's equations of motion and show that they are equivalent to your result for item (a).
A bead of mass m slides without friction along a rotating wire in the shape of...
A bead of mass m slides without friction along a rotating wire in the shape of a parabola with zar2, as shown below. The wire is rotating around the z-axis with constant angular velocity w z=ar2 (a) (0.5 point) Determine the Lagrangian for the system in terms of the coordinate r b) (1 point) Apply the Lagrange Equations to obtain the equation of motion. You (c) (0.5 points) Suppose that the bead is moving in a perfect circle of radius...
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variab...
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in equation (1) derive the trigonometric form of Legendre equation for a function T (0) where 0 θ π: sin θ Then the general solution to (3) is T (0) y(cos θ) AP, (cos0) + BQ, (cos0).
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in...
5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10)...
5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10) where 8, m and k are fixed real constants greater than zero. (total 10 points) (a) Write down the Euler-Lagrange equation of motion for this system, and interpret the resulting equation in terms of a known physical system. (1 point) (b) Find Hamiltonian via Legendre transformation. (1 point) (c) Show that q(t) and the corresponding canonical momentum p(t) can be found as follows for...
4. (Challenge Exercise) The parametric equations of a cycloid are: x = 2(0 - sin 8),...
4. (Challenge Exercise) The parametric equations of a cycloid are: x = 2(0 - sin 8), y = 2(1 - cos 6). Determine two points where the tangent to this cycloid is vertical.
2. Consider the parametric equation of a cycloid, given by r=r(@- sin()), y = r(1 -...
2. Consider the parametric equation of a cycloid, given by r=r(@- sin()), y = r(1 - cos(@)). (1) Find the equation of the tangent line to the cycloid at the point where @ = 7/4. (2) Find the area under one arch of the cycloid (0 SO2 ). (3) Find the length of one arch of the cycloid.
I need help with c). Question 2 The simple pendulum, discussed in week 4 and lab session 4, has the equation of motion...
I need help with c).
Question 2 The simple pendulum, discussed in week 4 and lab session 4, has the equation of motion f0/dt2_0? sin θ 9-(g/L)I/2. with The total energy of the pendulum is constant during the motion, and is given by _mgL cos θ, wher dt is the angular speed of the motion in radians per second. Consider the simple pendulum with initial conditions θ(0) and u(0)-wi, 0 i.e. starting from the vertically down position with an initial...
I need help with C Question 2 The simple pendulum, discussed in week 4 and lab session 4, has the equation of motion f0...
I need help with C
Question 2 The simple pendulum, discussed in week 4 and lab session 4, has the equation of motion f0/dt2_0? sin θ 9-(g/L)I/2. with The total energy of the pendulum is constant during the motion, and is given by _mgL cos θ, wher dt is the angular speed of the motion in radians per second. Consider the simple pendulum with initial conditions θ(0) and u(0)-wi, 0 i.e. starting from the vertically down position with an initial...
A frictionless wire is bent into the shape of a cycloid curve,
with coordinates given by the parametric equations
? = ?(? + sin ?), ? = ?(1 − cos ?), for −? < ? < ?. The x
axis is horizontal, and y is vertically upwards. A bead of
mass m slides freely on the wire. Show that the distance s,
measured along the wire from the origin, is given by
? = 4? sin. Write out the potential...
A bead of mass m slides along a frictionless wire under the influence of gravity. The shape of the wire is given by the equation y = axa, where x denotes the horizontal co-ordinate, y denotes the vertical co-ordinate, and a is a constant. (a) Use Lagrange's equation to determine the equation of motion. (b) Compute Hamilton's equations of motion and show that they are equivalent to your result for item (a).
A bead of mass m slides without friction along a rotating wire in the shape of a parabola with zar2, as shown below. The wire is rotating around the z-axis with constant angular velocity w z=ar2 (a) (0.5 point) Determine the Lagrangian for the system in terms of the coordinate r b) (1 point) Apply the Lagrange Equations to obtain the equation of motion. You (c) (0.5 points) Suppose that the bead is moving in a perfect circle of radius...
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in equation (1) derive the trigonometric form of Legendre equation for a function T (0) where 0 θ π: sin θ Then the general solution to (3) is T (0) y(cos θ) AP, (cos0) + BQ, (cos0).
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in...
5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10) where 8, m and k are fixed real constants greater than zero. (total 10 points) (a) Write down the Euler-Lagrange equation of motion for this system, and interpret the resulting equation in terms of a known physical system. (1 point) (b) Find Hamiltonian via Legendre transformation. (1 point) (c) Show that q(t) and the corresponding canonical momentum p(t) can be found as follows for...
4. (Challenge Exercise) The parametric equations of a cycloid are: x = 2(0 - sin 8), y = 2(1 - cos 6). Determine two points where the tangent to this cycloid is vertical.
2. Consider the parametric equation of a cycloid, given by r=r(@- sin()), y = r(1 - cos(@)). (1) Find the equation of the tangent line to the cycloid at the point where @ = 7/4. (2) Find the area under one arch of the cycloid (0 SO2 ). (3) Find the length of one arch of the cycloid.
I need help with c).
Question 2 The simple pendulum, discussed in week 4 and lab session 4, has the equation of motion f0/dt2_0? sin θ 9-(g/L)I/2. with The total energy of the pendulum is constant during the motion, and is given by _mgL cos θ, wher dt is the angular speed of the motion in radians per second. Consider the simple pendulum with initial conditions θ(0) and u(0)-wi, 0 i.e. starting from the vertically down position with an initial...
I need help with C
Question 2 The simple pendulum, discussed in week 4 and lab session 4, has the equation of motion f0/dt2_0? sin θ 9-(g/L)I/2. with The total energy of the pendulum is constant during the motion, and is given by _mgL cos θ, wher dt is the angular speed of the motion in radians per second. Consider the simple pendulum with initial conditions θ(0) and u(0)-wi, 0 i.e. starting from the vertically down position with an initial...
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