Problem 2 (40 Points) A particle of mass m is embedded at a distance a from the center of a massless circular disk of radius r. The disk rolls without slipping down a plane inclined at an angle a with the horizontal. A horizontal force of Ễ = −Fxî + Fyĵ resists motion of the disk down the plane by pushing on the disk at the axle that runs through the center of the disk. a) Find the kinetic energy T. (10 points) b) Find the potential energy V. (10 points) c) Write a position vector to the axle at the center of the wheel in terms of x and y. (10 points) d) Using virtual work, find the applied force Q₁ that would go in Lagrange's Equations. DO NOT WRITE OUT OR SOLVE LAGRANGES'S EQUATIONS. (10 points) x r m e 10 g F α HINTS 1) Consider using the STATIONARY red xy frame a reference frame from which to draw vectors 2) The red xy system DOES NOT move. It is stationary. 3) Consider that the disk rolls a distance of re down the ramp

4. A uniform rod is free to rotate about Point A. The tram that is holding A is accelerating to the left at 4 m/s². Neglect air resistance. When the rod is in equilibrium, at what angle, 0, will the rod be oriented relative to vertical? a = 4 m/s² A 0 = ? Rod g m = 6 kg L = 0.3 m

Please help This problem involves conservation of energy of rigid bodies. Thank you.

1. (bold type indicates a vector) An object of mass m = 10 kg may rotate in a vertical plane about the fixed point P under the action of gravity. The distance d from the point P to the center of mass C is d = 0.5 m. The moment of inertia Ie about the center of mass C is unknown. The object starts at rest in the position shown and is then released. At the instant of release the acceleration of the center of mass is measured to be a, = (-5 m/sec²)j (i.e., downward). Use the given information to determine the moment of inertià I, about the center of mass. Use g = 10 m/sec?. gravity w go P (f ixed)

The 2-mass system shown below depicts a disk which rotates about its center and has rotational moment of inertia Jo and radius r. The angular displacement of the disk is given by 0. The spring with constant k₂ is attached to the disk at a distance from the center. The mass m has linear displacement & and is subject to an external force u. When the system is at equilibrium, the spring forces due to k₁ and k₂ are zero. Neglect gravity and aerodynamic drag in this problem. You may assume the small angle approximation which implies (i) that the springs and dampers remain in their horizontal / vertical configurations and (ii) that the linear displacement d of a point on the edge of the disk can be approximated by d≈re. Ө K2 www m 4 Cz 777777 Jo Make the following assumptions when analyzing the forces and torques: тв 2 0>0, 0>0, x> > 0, >0 Derive the differential equations of motion for this dynamic system. Start by sketching LARGE and carefully drawn free-body-diagrams for the disk and the…

HW1 Four masses A, B, C and D are attached to a shaft and revolve in the same plane. The masses are 12 kg, 10 kg, 18 kg and 15 kg respectively and their radii of rotations are 40 mm, 50 mm, 60 mm and 30 mm. The angular position of the masses A, B, C and D are 60°, 90° and 70° from the mass A. Find the magnitude and position of the balancing mass at a radius of 100 mm.

A spring with a spring constant of 1200lbf/ft is attached to block with a mass of 96.6lbm horizontally. It was initially displaced 3 inches to the left of the equilibrium position and released at 12in/s going to the right. Determine also the c) acceleration 2 seconds after the push in ft/s2 *

2. Figure 2 shows a quick return mechanism. The ram at C has a mass of 50kg and is accelerating to the right at 10 m/s². In addition there is a force resisting the motion of the ram of 1500N. Determine the magnitude and direction of the forces: (a) acting between the ram and guide (b) at the pin joint B (c) (d) (e) Find the torque about O. at the pin joint Q. at the slider A. BL 45° 71° 18.5° Figure 2. Quick Return Mechanism OA = 0.075m AQ = 0.162m BQ = 0.1m BC = 0.3m

Question 1. A tube rotates in the horizontal ry plane with a constant angular velocity w about the z-axis. A particle of mass m is released from a radial distance R when the tube is in the position shown. This problem is based on problem 3.2 in the text. R m 2R Figure 1 x a) Draw a free body diagram of the particle if the tube is frictionless. b) Draw a free body diagram of the particle if the coefficient of friction between the sides of the tube and the particle is = k = p. c) For the case where the tube is frictionless, what is the radial speed at which the particle leaves the tube? d) For the case where there is friction, derive a differential equation that would allow you to solve for the radius of the particle as a function of time. I'm only looking for the differential equation. DO NOT solve it. 1 e) If there is no friction, what is the angle of the tube when the particle exits? • Hint: You may need to solve a differential equation for the last part. The "potentially useful…

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