Exercise 2: 1. Obtain the moment of inertia tensor of a thin uniform ring of radius R , and mass M, with the origin of the coordinate system placed at the center of the ring, and the ring lying in the xy plane. 2. Obtain the moment of inertia tensor of a uniform hollow sphere of radius R , and mass M , with the origin of the coordinate system placed at the center of the sphere. Note that this problem can be done by dividing the sphere into a large number of infinitesimally thin rings.

El péndulo, de masa \( m \) y largo \( L \), de la figura oscila de modo que la masa describe un movimiento circunferencial uniforme de radio \( R \), con una velocidad tangencial de magnitud \( v \).

A boat is supposed to travel with a velocity vector of \( \langle 40,30\rangle \), but the current has a velocity vector of \( \langle 0,-5\rangle \). What is the boat's actual velocity vector? \( \langle 45,30\rangle \) \( \langle 40,25\rangle \) \( \langle 40,35\rangle \)

14. A train moving at an average speed of \( 72 \mathrm{~km} / \mathrm{h} \) takes 15 seconds to completely cross a bridge that is 80 metres long (a) Express \( 72 \mathrm{~km} / \mathrm{h} \) in metres per second (b) Find the length of the train in metres ( 2 mk )

Question ix. An ideal gas is enclosed in a cylinder which has a movable piston. The gas is heated, resulting in an increase in temperature of the gas, and work is done by the gas on the piston so that the presure remains constant. a) Is the work done by the gas negative, positive or zero? b) From a microscopic view, how is the internal energy of the gas molecules affected? c) Is the heat less than, greater than or equal to the work? Explain you answer.

La cantidad de un material radiactivo disminuye a la mitad cada 10 años. Si comenzamos con 80 gramos, ¿cuánto quedará después de 30 años?