A straight, smooth tunnel is dug through a uniform spherical planet of mass density . The tunnel passes through the center of the planet and is perpendicular to the planet’s axis of rotation, which is fixed in space. The planet rotates with a constant angular speed ω such that objects in the tunnel have no apparent weight. What is the required angular speed ω of the planet?

Aplayground slide is 8.80 ft long and makes an angle of 25.0° with the horizontal. a 63.0-kg child, initially at the top, slides all the way down to the bottom of the slide. (a) choosing the bottom of the slide as the reference configuration, what is the system's potential energy when the child is at the top and at the bottom of the slide? what is the change in potential energy as the child slides from the top to the bottom of the slide? (include the sign of the value in your answer.)