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…

In an amusement park rocket ride, cars are suspended from 4.25-m cables attached to therotating arms at a distance of 6.00 m from the axis of rotation. The cables swing out at an angleof 45.0° when the ride is operating. Consider the mass of a rocket car is 75 kg. Find the radial and tangential acceleration of the rocket.

Two blocks m1=6.2kg and mɔ= 2.6kg are sliding on a circular track with radius R=5.6m as shown in the figure below. If the two blocks are moving together without m2 sliding off of m, and their speed is v=11.9m/s, determine the normal force applied by the track to m, at the instant shown in the figure when the angle 0=41°. Express your answer in units of Newtons (N) using zero decimal places. Hint : Ignore the size of the blocks and take g=9.80 m/s?. Zu m1