Steiner’s theorem and gyration radius. A metal tank wheel in the shape of a disc of radius R is lightened by cutting out six identical through- holes in the shape of discs, whose axes are parallel to the wheel’s axis of rotation, and whose radii are all r. The holes’ centers are positioned at the vertices of a regular hexagon of side A, with the wheel’s axis passing through the center of the hexagon. a) By what percentage is the wheel lightened, as a result of the cutouts, for r=R/5?

A flywheel with a radius of 0.320 m starts from rest and accelerates with a constant angular acceleration of 0.710 rad/s². Part A Compute the magnitude of the tangential acceleration, the radial acceleration, and the resultant acceleration of a point on its rim at the start. Enter your answers separated with commas. atan, arad, atotal = Submit Part B ΠΑΣΦ Request Answer atan, arad, atotal = Compute the magnitude of the tangential acceleration, the radial acceleration, and the resultant acceleration of a point on its rim after it has turned through 60.0 °. Enter your answers separated with commas. ΑΣΦ m/s² ? m/s²

A a paddle for a boat is constructed of a stick of length L and a "quartic" end piece. The end can be thought of as a uniform flat board of mass m that has been cut to form a quartic function given by y = x*, so that the length from the trough (i.e., the y 0 end of the function) to the tip is y = H. To simplify this a little, assume that the stick is massless. This paddle is stood vertically upward on the stick end and attached to a hinge on the floor so that it is free to rotate in the in/out of the page direction. When it is standing straight up that is an unstable equilibrium. A very small infinitesimal push is given to get the paddle moving and it swings downward under gravity (acceleration g). (a) When the paddle is standing vertically upward, what is the position of the center of mass of the paddle? (b) What is the moment of inertia of the paddle for rotations in/out of the page about an axis through the center of mass? (c) What is the angular velocity of the paddle just before…

I got Part A really well in finding the torque but I am struggling to find the angle theta in part B.

A В C A cylindrical object rolls without slipping down a ramp and around a loop-the-loop as shown in the figure above. The object maintains contact with the ramp and loop at all times. At position B, (just after passing the top part of the loop) what direction is the angular acceleration of the cylinder? Up and to the right Down and to the left Down and to the right Towards the center of the circle Downward Into the page Out of the page