Two uniform, solid cylinders of radius R and total mass M are connectedalong their common axis by a short, light rod and rest on a horizontaltabletop. A frictionless ring at the rod’s center is attached toa spring of force constant k; the spring’s other end is fixed. The cylindersare pulled to the left a distance x, stretching the spring, then releasedfrom rest. Due to friction between the tabletop and the cylinders, the cylindersroll without slipping as they oscillate. Show that the motion of thecenter of mass of the cylinders is simple harmonic, and find its period.

THE MOTION IS ON THE HORIZONTAL PLANE (NO GRAVITY

Question 2: A point-mass M of mass m is tied to a string which winds around a cylinder like a bobbin. The string is pinned on the cylinder at some point, and P denotes the point at which the string leaves the bobbin and starting from which it is taut. The cylinder has radius a and rotates at a constant rate around the vertical axis of the fixed frame F (which is also the axis of symmetry of the bobbin). The motion takes place in the hori- zontal plane, which is assumed frictionless. See figure. (a) Find a relationship between l, a and expressing that the length of the string is constant (b) Compute the velocity and acceleration of point M with respect to F. (c) Apply LMB in F to the system consisting of the point mass M alone (don't forget to start with a free body diagram!). Deduce a differential equation for l as well as the value of the string's tension in terms of the other parameters. (d) (bonus) Solve the differential equation for l. eo Ω Mas P M FIGURE 1 - The spinning bobbin.…

The mechanism shown consists of a crank (bar AB), a connecting rod (bar BC) and the piston C that slides on the smooth surface (without friction). The combustion of gasoline produces a force P on the piston and this is kept in equilibrium with the moment M applied in A. The length of the crank is 116 mm, the length of the connecting rod is 183 mm, the force P is 710 N and the angle theta is 0.56 radians. Determine the value of M in Nm.

5. A uniform solid cylinder with M 2R mass M and radius 2R rests on a horizontal tabletop. A string is attached by a yoke to a frictionless axle through the center of the cylinder so that the cylinder can rotate about the axle. The string runs over a disk-shaped pulley M with mass M and radius R that is mounted on a frictionless axle through its center. A block of mass M is suspended from the free end of the string. The string doesn't slip over the pulley surface, and the cylinder rolls without slipping on the tabletop. Find the magnitude of the acceleration of the block after the system is released from rest.

7. A cage of mass 2500 kg is raised and lowered by a winding drum of 1.5 m diameter. A brake drum is attached to the winding drum and the combined mass of the drums is 1000 kg and their radius of gyration is 1.2 m. The maximum speed of descent is 6 m/s and when descending at this speed, the brake must be capable of stopping the load in 6 m. Find 1. the tension of the rope during stopping at the above rate, 2. the friction torque necessary at the brake, neglecting the inertia of the rope, and 3. In a descent of 30 m, the load starts from rest and falls freely until its speed is 6 m/s. The brake is then applied and the speed is kept constant at 6 m/s until the load is 10 m from the bottom. The brake is then tightened so as to give uniform retardation, and the load is brought to rest at the bottom. Find the total time of descent. [Ans. 32 kN ; 29.78 kN-m ; 7.27 s]

A string is wrapped around a uniform disk of mass M = 2.1 kg and radius R = 0.12 m. (Recall that the moment of inertia of a uniform disk is (1/2) MR².) Attached to the disk are four low-mass rods of radius b = 0.16 m, each with a small mass m = 0.7 kg at the end. The device is initially at rest on a nearly frictionless surface. Then you pull the string with a constant force F = 30 N. At the instant when the center of the disk has moved a distance d = 0.041 m, a length w = 0.033 m of string has unwound off the disk. •m bi m m Part 1 M, R Part 2 d V= 0.708548386 Your answer is correct. Part 3 m (a) At this instant, what is the speed of the center of the apparatus? @01 = i 5.08017 Save for Later F Your answer is incorrect. (b) At this instant, what is the angular speed of the apparatus? 002 = i w+d Save for Later m/s eTextbook and Media F radians/s (c) You keep pulling with constant force 30 N for an additional 0.042 s. Now what is the angular speed of the apparatus? radians/s Attempts: 1…

A circular rigid disk of mass m kg and radius rm is stationary at the top of a fixed sphere of radius 2r m. The disk is disturbed slightly from rest and rolls with planar motion down the sphere without slipping. Determine the value of angle at which the disk leaves the sphere. Derive the tangential friction force F between the disk and sphere as a function of angle 0. sphere -2r. disk o'r

Please show all work and diagrams! Anything is helpful.

1. Two identical beads A and B each of mass m connected by a light rod of length rv2 can slide without friction on a fixed ring of radius r. The beads are also connected to fixed points C and D with identical elastic cords each of force constant k and relaxed length rv2 as shown in the figure. Initially the cords coincide with diameters of the ring and the system is in equilibrium. Now the bead-rod assembly is displaced slightly aside and released. Find acceleration of the point of intersection Pof the cords immediately before a cord becomes relaxed. B D Ans = I need to know how this answer mv2 Comes show every step of solution P.