question (c), (d), (e), (f) please. Thanks.
question (c), (d), (e), (f) please. Thanks. 1 Consider a cylinder of mass M and radius...
question (c), (d), (e), (f) please. Thanks. 1 Consider a cylinder of mass M and radius a rolling down a half-cylinder of radius R as shown in the diagram (a) Construct two equations for the constraints: i rolling without slipping (using the two angles and θ), and ii) staying in contact (using a, R and the distance between the axes of the cylinders r). (b) Construct the Lagrangian of the system in terms of θ1, θ2 and r and two...
1 Q2. Figure 2 shows a system in which mass m is connected with a cylinder of mass m2 and moment of inertia Jo through a horizontal spring k. The cylinder is m1 rolling on the rough surface without slipping. (1) Find its total kinetic energy, total potential energy TN and Lagrangian, Figure 2 (2) Derive the equations of motion using Lagrangian equation method, and (3) Calculate its natural frequencies 1 Q2. Figure 2 shows a system in which mass...
ANS: PLEASE USE LAGRANGIAN, THANK YOU, WILL UPVOTE GOOD ANSWER IMMEDIATELY Consider a uniform disk of mass m, and radius R that is rolling with slipping. The surface has a coefficient of kinetic friction a) Find the equations of motion. b) Next consider the same disk when it is rolling without slipping. Find the EOM using either x or θ. Hint: be careful with the generalized force for θ. If we label point P as the point on the disk...
Q3. For the system in Figure 3 where and θ2 are the rotational angles, and are the rotary inertias of the two disks with radius r and 2r, respectively, 2r (1) Find its total kinetic energy, total potential energy and Lagrangian in terms of, and (2) Derive the equations of motion using Lagrangian equation method, (3) Put the equations of motion in matrix form, and (4) Calculate the natural frequencies and the associated mode shapes if m-30 g, 4-8 x...
6. (Bonus question.) A small uniform cylinder of radius R rolls without slipping along the inside of a large, fixed cylinder of radius r > R as shown in the figure above. mig (a) Use conservation of energy to show that the period of small oscillations (θ « 1) of the rolling cylinder about the equilibrium position O is equivalent to that of a simple pendulum whose length is (r-R). [Note that the rotation rate w of the cylinder is...
4) Figures 4A (side view) and 4B (overhead view) illustrates a uniform solid cylinder having mass M and radius R. The cylinder is positioned on a horizontal floor having sufficient friction to ensure that the cylinder can roll without slipping. The cylinder includes a mass-less yoke that is fixed to the symmetric axis of the cylinder and acts as a rolling friction-less pivot for the cylinder. An ideal spring having spring constant K is attached to the yoke at one...
A small sphere of mass m and radius r is placed inside a horizontal hollow cylinders of radius Now cylinder rotating with some angular acceleration which slowly increases from zero to certain value There is no slipping between two surfaces during motion:- (A) Maximum angle formed by line joining centre will be 0 =sin 2aR 5g (B) Angular velocity of sphere will be m'= where is angular velocity of cylinder R (C) Sphere will keep rotating at lowest position (D)...
The axis of a smooth fixed ciular cylinder of radius R is horizontal. A particle of mass m is attached to a model string and is initially at rest level with the cre of the cylinder with the string draped over the top, whee t slids without frictian as if on a model pulley, s shown in the diagram below. A constant force P of magnitude P pulls the model string downwards. Let 0 denote the angle subtended at the...
Bridging Problem: Oscillating and Rolling Two uniform, solid cylinders of radius R and total mass M are connected along their common axis by a short, light rod and rest on a horizontal tabletop (Figure 1). A frictionless ring at the rod's center is attached to a spring of force constant k, the spring's other end is fixed. The cylinders are pulled to the left a distance I, stretching the spring, then released from rest. Due to friction between the tabletop...
A light cable is wrapped around a solid cylinder with mass M = 1.50 kg and radius R = 0.20 m as shown in the figure below. The cylinder rotates with negligible friction about a stationary horizontal axis. The free end of the cable is tied to a block of mass m = 0.50 kg. The block, initially at rest, is released at a height h = 1.00 m above the floor. As the block falls, the cable unwinds without...