A wheel is fixed at its center position and allowed to rotate. The moment of inertia of the wheel is J, and the wheel is connected to the fixed center point through a torsional spring with spring constant k(subscript 1) (not shown in the above figure). This wheel is in contact with mass M. The wheel and the mass maintain rolling without sliding condition at the contacting surface. We apply a torque T at the wheel. Derive the Equation of Motion for the system. You should have only one unknown response in your equation. Use x(t) . I am having a lot of trouble with this problem - please be very detailed and explain fully!

Please solve only 4,5

Problem 2: A car is pushing a barrel to the right. The pushing force P is applied by the front bumper which is level with the center G of the barrel (see figure). The barrel has mass m, radius R, and radius of gyration kG. The coefficients of static and kinetic friction between the bumper and the barrel are µs and µk. The coefficients of static and kinetic friction between the barrel and the road are also µs and µk. 1) If the barrel doesn't slip on the road, find the acceleration of the barrel and the minimum value of us required to ensure no slipping. 2) If the barrel slips on the road, find the acceleration of G and the angular acceleration of the barrel. Hs. Hk. m

The structure of a football goal is shown in the figure below. The bending motion of the structure can be represented by a flexible cantilever beam with tip mass M¡ as shown in the figure. The tip mass M, represents the mass of the horizontal crossbar which is assumed to be rigid in this motion. A single vertical straight beam having mass of MH, shown in the figure below, is the equivalent model for the two symmetrical vertical posts. A soccer ball of mass M, hits the horizontal crossbar with the velocity of Vp. x(t) м, VINGLI SOLAR The response of the horizontal crossbar, mass M, is shown by x(t) in the figure. The vertical beam has a circular cross section of radius r, Young's modulus E and length L. For numerical calculations, use M, = 23.2130 kg, MH = 28.6617 kg. M, = 0.5497 kg, Vý = 8.8974 m/s, L= 2.7757 m, E = 133.5685 GPa andr= 0.0437 m. For circular cross- section I = Tr/4.Considering zero damping and the numerical data given above, obtain the maximum amplitude of response x(t)…

(3) A circular cam 8-in in diameter rotates off-center with an eccentrity e = 1" & operates the roller follower that is carried by the arm. The roller follower is held against the cam by means of a tension spring attached a third from a fixed pivot point. The force against the cam & roller follower is approximately 50lb @ the low position & 80lb @ high position. The ratio of the coil diameter to that of wire diameter is 7.0. The allowable stress on the spring is S,= 60,000psi. Calculate : P (a) the wire diameter; (b) the outside diameter of spring; (c) the number of active coils recommended.

please do it neatly and correct and fast please

A turbine rotor is mounted on a stepped shaft that is fixed at both ends as shown in The torsional stiffnesses of the two segments of the shaft are given by ka = 3,000 N-m/rad and k2 = 4,000 N-m/rad. The turbine generates a harmonic torque given by M(t) = Mo cos wt about the shaft axis with M, = 200 N-m and w = 500 rads. The mass moment of inertia of the rotor about the shaft axis is Jo = 0.05 kg-m. Assuming the equivalent torsional damping constant of the system as c, = 2.5 N-m-s/rad, determine the steady-state response of the rotor, 6(1). O(1) ke M(1) = M, cos ot Turbine rotor, Jo

7. 2) The lawn roller has a mass m = 55 kg and a radius of gyration about its gravitational center G, kG = 196 mm. It has a handle that connects to its center axle, and the handle is pushed forward with a force F = 420 N at a 45° angle. The coefficients of static and kinetic friction between the roller and the ground are us = 0.11 and Hk = 0.08, respectively. For the sets of parameters listed below, determine the magnitude of the roller's angular acceleration (in rad/s²). Please pay attention: the numbers may change since they are randomized. Your answer must include 2 places after the decimal point. Take g = 9.81 m/s?. F 45° 200 mm A

For two existing torques, what third force at a given distance from the pivot will balance them? Imagine a meter stick set up as in the figure. It hangs from a central bracket, and two hanging masses can hang from it from each of their brackets. At a third location, a force probe can either pull up or pull down on the stick, depending on what is needed to balance the stick. The mass of the meter stick is 120 g. sketch the situation (drawing r1, r2, r3, F1, F2, and F3) and determine the magnitude (value) and direction (+ or -) of each torque. Don't include the mass of a bracket that would hold the hanging mass in place; assume the mass listed is the entire mass hanging at that point. For each trial, use the principle of equilibrium (where the sum of torques is zero) to calculate the third, unknown force acting at x3

A pendulum is made up of a spring with a mass attached to the end. The spring is arranged to lie in a straight line, by wrapping the spring around a massless bar. The equilibrium length of the bar is “l”. Using lagrange’s equations, find the equations of motion.