The figure below shows particles 1 and 2, each of mass m, attached to the ends of a rigid massless rod of length L1 + L2, with L = 22 cm and L2 = 84 cm. The rod is held horizontally on the fulcrum and then released. (a) What is the magnitude of the initial acceleration of particle 1? m/s2 (b) What is the magnitude of the initial acceleration of particle 2? m/s²

A bead of mass m slides without friction along a curved wire with shape z = f(r) where r = Vr2 + y², i.e. the distance from the z-axis. The wire is rotated around the z-axis at a constant angular velocity w. Gravity acts downward along the z-axis with a constant acceleration g. a) Using Newton's second law in an inertial frame, derive an expression for radius ro of a fixed circular orbit (i.e. a solution with r = ro = const.). What is the normal force the wire applies to the bead to keep it in a circular orbit? b) Show that the equation of motion for r(t) (general equation not the circular motion) is F(1 + f'(r)²) + i² f'(r)f"(r) + gf'(r) – w?r = 0. Using this verify your answer to part (a). c) Consider small displacements from the circular orbit, r = ro+e(t). Derive a condition on the function f(r) such that a circular orbit at r = ro is stable. d) Find the force on the bead in the o direction, i.e. perpendicular to the plane of wire. The angular velocity is w = . Obtain the answer…

A tire 1/2 m in radius rotates at a constant rate of 240 rev/min. Use v = rw to find the linear speed in meters/sec of a small stone lodged in the tread of the tire (on its outer edge).

Two people swing jump ropes, as shown in the diagram. The highest point of the middle of each rope is M = 77 inches above the ground, and the lowest point is m 7 inches. The rope makes 4 revolutions per second. Write a model for the height h (in inches) of a rope as a function of the time t (in seconds) given that the rope is at its lowest point when t = 0. i ↑ h = h(t) = M Kit m -