Question

Consider a rod of length L and mass M. To approximate the rotational inertia of this stick, we will imagine it divided into five equal pieces. Consider all of the mass of each piece (M/5) to be located at the middle of each piece. Thus, mass locations will be 0.1L, 0.3L, etc. a) Determine the rotational inertia of the rod by summing the contributions of each of the five mass pieces. b) Compare this to that given (in your textbook, for example) for a rod rotating about its end.

Answer 1

a) I = 0.33 ML² , b) I_exact / I = 1.01

Step-by-step explanation:

a) In this case we are asked to find the moment of inertia of a bar, assuming that it is made up of equally separate point masses.

The bar is divided into 5 parts of equal mass, therefore each one has a mass of m / 5 and the positions of the 5 point masses are

0.1L, 0.3L, 0.5L, 0.7L, 0.9L

the formula for the moment of inertia of a point masses are

I = ∑ M R²

let's replace

I = M / 5 [(0.1L)² + (0.3L)² + (0.5L)² + (0.7L)² + (0.9L)² ]

I = M L²/5 [1.65] ²

I = 0.33 ML²

b) The theoretical value that we can obtain using integrals is given for a bar with the axis of rotation at one end by

I_exact = ⅓ ML²

I _exact = 0.33333333 M L²

the difference between the two quantities can be found by the ratio between them

I_exact / I = 0.33333333333 / 0.33

I_exact / I = 1.01

we see that the difference is 1% so this calculation is quite good