Suppose you're eating in yet another restaurant where the dishes are shared at the table and all

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Question:

Suppose you're eating in yet another restaurant where the dishes are shared at the table and all placed uniformly on a rotating disk-like surface. Model this surface as a thin disk of radius 0.4 cm. Someone else has spun the surface, such that it is initially at an angular speed of 0.3 rev/s. The surface and food has a combined mass of 4.4 kg. The waiter, to show off, throws a new dish of dumplings (mass 1 kg) onto the surface at a speed of 0.8 m/s, such that the dish lands on and sticks to the very edge of the surface moving in the same direction as the rotating food. While this is happening, you quickly calculate the final angular speed of the food so that you can predict its location at any time before others have a chance to eat the dumplings. What is this speed, in rad/s?

Suppose you observe your friend (mass 69.1 kg) stand at the very edge of a motionless merry-go-round (modelled as a thin disk with moment of inertia 6.8 kg m 2). At some point, your friend walks around and along the edge at a speed 1.1 m/s perfectly tangent to the edge at all times, relative to the merry-go-round's surface. Take the radius of the merry-go-round to be 1.4 m.

Calculate the speed (in m/s) of your friend relative to the stationary ground.

Suppose you're eating in a restaurant where the dishes are shared at the table and all placed uniformly on a rotating disk-like surface. Model this surface as a thin disk of radius 40.9 cm. You can't stop thinking about physics even though you're out with your friends and decide to calculate the mass of the rotating surface and all the food. If the surface is initially at rest and you exert a tangential force of 2.3 N on it, you observe that the food rotates at a speed of 1 rev/s after applying the force consistently for 1.5 seconds. Find the mass of the disk with the food, in kg.

Suppose a friend is completing another physics assignment in their chair at their desk and you take the opportunity to turn this scenario into the physics question you now must complete. Your friend and chair have a combined constant moment of inertia of 5.4 kg m2, and the chair can freely rotate. Now your friend picks up two solid bricks (each of mass 1.9 kg) they accidentally collected from one of the campus buildings, and experiment with rotational kinematics. Supposing they hold the masses in each hand, arms outstretched, at distance of 0.8 m from their axis of rotation, and have an initial angular speed of 2.6 rev/s.

(a) Calculate the distance in m of each brick from your friend's axis of rotation if they pull the bricks inward such that their rotational speed has increased to 3.1 rev/s.

(b) What is the initial rotational kinetic energy of the system?