Question

Consider a thin rod of length L and mass M situated with one end at the origin x = 0 of the coordinate system as shown

a) Four 1 kg boxes sit on a 5 m long uniform rod of mass 4 kg, such that the total mass of the system is 8 kg. The boxes are spaced 1 m apart, with the first box sitting at the left end of the rod (i.c. at i-om). x-Om x-5 m i) Please show mathematically that the center of mass of the system (i.e. the four boxes plus the uniform rod) is at 1-2 m. (Note: you may treat each of the boxes as a point mass.) ii) If the fourth block (located at x = 3 m) is removed, does the center of mass of the system move to the left, move to the right, or stay at the same location? Please explain. b) Consider a thin rod of length and mass situated with one end at the origin -0 of the coordinate system as shown. i) If the rod has a uniform mass distribution, please prove (using integration) that the moment of inertia of this rod about the originis ML x-o x-1 ii) If instead the rod has a linearly increasing mass distribution (e.g. 1. -ax, like the baseball bat done in class), do you expect the moment of inertia about the origin to increase, decrease, or remain the same (compared to F.ML?)? Please explain.

Mostly need help with part B. Thank you

Answer 1

L dx X of Let us consider a small elemento thickness dx at distance & from axis of rotation ^ dm= M dx 2 As dl= dim ne? dl= M x²du I- M L² 3

is linearly more (in if the mass distribution then mass will be concentrated away from axis I= dm x² x increases for AS larger dm. for larger overal effect is that it increases I. Moment of inertia increases iii) is more same If mass concentrated near mid point, we will expect to have moment of inertia. As inija V centre of mass was mid point of rod and all was can be viewed to be concentrated at mid point Rema in the same