A ball of mass m, slides over a ếy sliding inclined plane of mass M and angle a. Denote by X, the coordinate of O' with re- spect to 0, and by (x.y) the coordinate of m with respect to O'(see figure below) m M a O' 1. Calculate the degree of freedom of the system 2. Find the velocity of m with respect to 0. 3. Write the expression of the Lagrangian function
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- Problem 3 A ball of mass m, slides over a ēy sliding inclined plane of mass M and angle a. Denote by X, the coordinate of O' with re spect to 0, and by (x.y) the Coordinate of m with respect to O'(see figure below) 1. m 1. Calculate the degree of freedom of the system 2. Find the velocity of m with respect to O. 3. Write the expression of the Lagrangian function 4. Derive the Euler Lagrange equations 5. Find z" and X" in terms of the masses (m.M), angle a and gProblem 3 A ball of mass m, slides over a ey sliding inclined plane of mass M and angle a. Denote by X, the coordinate of O' with re- spect to 0, and by (x.y) the coordinate of m with respect to O'(see figure below) M a O' 1. Calculate the degree of freedom of the system 2. Find the velocity of m with respect to O. 3. Write the expression of the Lagrangian function 4. Derive the Euler Lagrange equations 5. Find z" and X" in terms of the masses (m,M), angle a and gLagrangian Dynamics O' A simple pendulum of mass m is piv- oted to the block of mass M, which slides on a smooth horizontal plane. O The mass M is connected to a spring of stiffness k, and the reference of poten- tial energy is at the smooth horizontal plane. ee 1. Find the velocity of mass m, w.r.t the origin O 2. Write the Lagrangian of the system 3. Derive the Euler Lagrangee equations 3.
- Problem 3 A ball of mass m, slides over a sliding inclined plane of mass M and angle a. Denote by X, the coordinate of O' with re- m spect to 0, and by (x.y) the Coordinate of m with respect M a O' to O'(see figure below) 1. Calculate the degree of freedom of the system 2. Find the velocity of m with respect to O. 3. Write the expression of the Lagrangian function 4. Derive the Euler Lagrange equations 5. Find z" and X" in ters of the masses (m,M), angle a and gLagrangian Dynamics X O' M A simple pendulum of mass m is piv- oted to the block of mass M, which slides on a smooth horizontal plane. O The mass M is connected to a spring of stiffness k, and the reference of poten- ee tial energy is at the smooth horizontal plane. 1. Find the velocity of mass m, w.r.t the origin O 2. Write the Lagrangian of the system 3. Derive the Euler Lagrange equationsA particle is moving on top of a 2-dimensional. plane with its coordinates given incartesian system asx(t) = a sin ωt, y(t) = a cos ωt.Express the motion of the particle in terms of polar coordinates (ρ, φ). What is the minimum numberof generalised coordinates required to describe. its motion? Draw the. trajectory of the particle.Now if the particle trajectory is changed to the followings, repeat the exercise.x(t) = 2a sin ωt, y(t) = a cos 2ωt
- A particle of mass m is projected upward with a velocity vo at an angle a to the horizontal in the uniform gravitational field of earth. Ignore air resistance and take the potential energy U at y=0 as 0. Using the cartesian coordinate system answer the following questions: a. find the Lagrangian in terms of x and y and identiy cyclic coordinates. b. find the conjugate momenta, identify them and discuss which are conserved and why c. using the lagrange's equations, find the x and y components of the velocity as the functions of time.A BO Two objects with identical masses m are tied to each other with a string of length L. The figure shows the situation at the initial time t = 0. At that instant, the string is tight, B is at rest, A is moving with velocity i A = i - vo . We will call the reference frame in which the problem is stated as the lab frame. It is easier to investigate the motion and answer the following questions in the center of mass frame. For this reason, first we express the initial conditions in this frame. (a) Find the initial velocity of A and B relative to the center of mass frame. Note: you should enter the unit vectors i,j, k as ihat, jhat and khat. %3D (b) Find the tension in the string Tension = (C) Find the period of the motion in the center of mass frame. tperiod (d) Suppose that half a period has passed, i.e, the time is nowt = tperiod/2. Find the velocities of A and B in the lab frame at that time. %3DDefine ond nlustrate general collis fons of two objects two- dimensional space. Please define the conservation of total momentun and energy of the system în each direction.
- Problem 3 A ball of mass m, slides over a sliding inclined plane of mass M and angle a. Denote by X, the coordinate of O' with re- spect to 0, and by (x.y) the Coordinate of m with respect to O'(see figure below) M O' 1. Calculate the degree of freedom of the system 2. Find the velocity of m with respect to O. 3. Write the expression of the Lagrangian function 4. Derive the Euler Lagrange equations 5. Find z" and X" in terms of the masses (m,M), angle a and gIn a clamped frictionless pipe elbow (radius R) glides a sphere (weight W = mg) with zero initial velocity downwards from the top. %3D Determine the support reactions at the Clamping (wall connection) in dependence on the position o of the sphere. At which o the reactions take extreme values?Give at least three (3) real life situation that demonstrate Lenz's law. Explain.