Using Lagrangian formalism, solve the following problem: A solid cylinder is released from rest to roll without slipping down a ramp of slope ϕ. a) What is the best choice of generalized coordinates to be used?
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- Write down the inertia tensor for a square plate of side ? and mass ? for a coordinate system with origin at the center of the plate, the z-axis being normal to the plate, and the x- and y- axes parallel to the edges.Please obtain the same result as in the book.For a particle of mass m and situated at position r from the origin, show that the moment fo force(torque) is given by: torque = dL / dt where L is the angular momentum Hence, show that for a conservative system, L is constant.
- Engineering Dynamics need help from 4,5,6,7 thank you A ball of mass m is moving along a vertical semi-cylinder of radius R as it is guided by the arm OA. The arm moves in a clockwise direction with a constant angular velocity ω. Assume 0° ≤ Φ ≤ 90°. Neglect any friction. Neglect also the size of the ball and the thickness of the arm. Find the relationship between r, R and θ where r is the distance between O and the ball. Draw a free body diagram of the ball assuming that it is in contact with the cylinder and the arm OA. Write the equations of motion in the (r, θ) coordinate system. Find the normal force acting on the ball by the cylinder for Φ = Φ0. Find the normal force acting on the ball by the bar for Φ = Φ0. Determine the angle Φ at which the ball loses contact with the cylinder. Take m = 1 kg, R = 1.4 m, ω = 0.5 rad/s, and Φ = 60°A disk of mass M and radius R rolls without slipping down a fixed inclined plane that makes an angle a with the horizontal plane. (see figure). RPO P.E = 0 (a) Write down the constraint equations and determine the number of degrees of freedom s. (b) Choose a convenient generalized coordinate. (c) Write down an expression for the Lagrangian of rolling ball. Given: the moment of inertia of the ball about its center is I, = }MR² (d) Calculate the generalized momenta.A uniform rod of mass M and length 2a lies along the intervals (-a, a] of the x-axis and a particle of mass m is situated at the point x = x'. Show all work in this problem, even if you happen to have written this particular solution on your formula sheet (you must rederive it). (a) Find the gravitational force exerted by the rod on the particle. (b) What would happen if the particle were replaced by a hollow sphere of the same mass, centered at x'. Does your answer depend on how large the sphere is?
- Consider the schematic of the single pendulum. M The kinetic energy T and potential energy V may be written as: T = ²m²²8² V = -gml cos (0) аас dt 80 The Lagrangian L is given by L=T-V, and the Euler-Lagrange equations for the motion of the pendulum are given by the following second order differential equation in : ас 80 = 11 = 0 Write down the second order ODE using the specific T and V defined above. Please write this ODE in the form = f(0,0). Notice that this ODE is not linear! Now you may assume that l = m = g = 1 for the remainder of the problem. You may still suspend variables to get a system of two first order (nonlinear) ODEs by writing the ODE as: w = f(0,w) What are the fixed points of this system where all derivatives are zero? Write down the linearized equations in a neighborhood of each fixed point and determine the linear stability. You may formally linearize the nonlinear ODE or you may use a small angle approximation for sin(0); the two approaches are equivalent.Angular momentum plays a key role in dealing with central forces because it is constant over time. Suppose the angular momentum, L, of a point mass is given by: By nature of the cross product, what two properties does L have? Show that: If a particle is subject to a central force only, then its angular momentum is conserved i.e. If V(r) = V(r), then dL/dt = 0.Show that the system of momenta for a rigid body in plane motion reduces to a single vector, and express the distance from the mass center G to the line of action of this vector in terms of the centroidal radius of gyration k of the body, the magnitude v of the velocity of G, and the angular velocity w.