3. A slender, rigid bar of mass 20 kg and length 4 m is rotating about its pin joint at Point A as shown in the figure below. A 50-N force is applied to the end of the rod in the vertical direction. Determine the reactions at the pin joint as well as the change in angular rotation rate at the instant shown. Use 0 = 30° and 8 = 2 rad/s. e P

Determine the reactions at the pin support and the applied moment if the bar of figure (shown below) has a mass of 50 g. If θ is 15° and the angular acceleration is 20 rad/s².

4. A uniform disc of mass m = 3 kg and radius R= 0.5 may roll without slip on the 30 degree incline shown. An elastic spring of stiffness k= 100 newtons/meter is connected to the center C of the disc. Initially, the disc is held in place in the position shown, with the spring neither stretched nor compressed and with the disc at rest. At t = 0 the disc is released and is subjected to a constant force of Fo = 30 newtons along the incline. Determine the maximum speed achieved by the disc center C as the disc moves up the incline. For gravity use g = 10 m/sec². (note: initially the constant force Fo will be greater than the forces exerted by the spring and by gravity, and the disc will accelerate up the incline; but there will come a distance x along the incline after which the spring force + gravity will overtake the force Fo, and the disc will then decelerate). gravity Fo= 30 N 30° no friction

In the figure below, the disc has mass m = 82.2998 kg, radius r= 6.7128 m, and moment of inertia about the mass centre l= mr2/2, and is attached to a block of mass M= 153.7102 kg which rolls across the surface. The block is supported with a spring with stiffness K = 630589.8842 N/m. If the disc rolls without slip and the block moves without friction, determine the natural frequency of the system.