Given: The gear shown moves on a fixed horizontal rack. The gear has mass m and a radius of gyration of k equal to its radius. The rod BC connected to the gear has length 1, and mass 2m with center of mass located at pt. G. The slider at B has negligible mass, and the attached spring has a spring constant k equal to mg/l. The system is released from rest in a vertical position (the bar BC is vertical). In this position, the spring is unstretched. The system has negligible friction and mass of the spring. B mrod G R mrod = 2m mgear = m 4 I mgear, k Find: Find the velocity of the point C on the gear after it has moved a distance of 1/2 to the right.

Problem 5: A wheel is released at the top of a hill. It has a mass of 150 kg, a radius of 1.25 m, and a radius of gyration of kg -0.6 m. be a). Assume that the wheel does not slip as it rolls down the hill. Derive the equation of motion using Lagrange equations. b). Assume the wheel does slip, and that the friction force between the wheel and the ground is 7 = -3Ni. Derive the equations of motion using Lagrange equations. Concept question: For which case, a) or b) will friction appear in the equations of motion as a non-conservative generalized force? A. in a) only, B. in b). only, C. in a) and b), D. Never.

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A shaft turning at a uniform speed carries two uniform discs A and B of masses 10kg and 8kg respectively.  The centres of the mass of the discs are each 2.5mm from the axis of rotation.  The radii to the centres of mass are at right angles.  The shaft is carried in bearings C and D between A and B such that AC = 0.3m, AD = 0.9m and AB = 1.2m. It is required to make dynamic loading on the bearings equal and a minimum for any given shaft speed by adding a mass at a radius 25mm in a plane E. Determine: (a) The magnitude of the mass in plane E and its angular position relative to the mass in plane A (b) The distance of the plane E from plane A (c) The dynamic loading on each bearing when the mass in plane E has been attached and the shaft rotates at 200 rev/min.  For the bearing loads in the opposite direction determine all the unknown values.  For the bearing loads in the same direction, show the diagrams and equations only to use for a possible solution.

A four-wheeled vehicle of total mass 5000 kg has two pairs of wheels of 500 mm radius. Each pair of wheels with axles has mass of 750 kg, the radius of gyration of each wheel being 400 mm. The axles are 2 metres apart and the c.g. of vehicle is mid-way at a height of 1-2 m above the road surface. Find the reactions of the front and rear wheels, if tractive force of 6 kN is acting 50 mm below the centre of gravity of vehicle.

A drum can rotate about a fixed-point O. The A block is attached to a cord wrapping around the drum. The mass of the drum is md = 100kg and the radius is r = 0.5 m. The radius of gyration of the drum about point O is ko=0.3 m. The mass of the block is mb= 20kg. The block is released from rest. The acceleration due to gravity is g=9.81 m/s2.   (1) Calculate the mass moment of inertia of the drum about the point O, IO_______(kgm2) (two decimal places)