# Angular momentum and moment of intertia

Not what you're looking for?

See attached file

##### Purchase this Solution

##### Solution Summary

A detailed solution is given.

##### Solution Preview

1) The position of the center of mass for a system of point masses located at positions r_{i} and masses m_{i} is by definition:

R = 1/M Sum_{i} m_{i} r_{i} (1.1)

where M is the total mass:

M = Sum_{i} m_{i} (1.2)

We want to prove that this formula also holds for a collection of extended masses if you take the r_{i} to be the locations of the center of masses. If you reorder the summation in (1.1) so that you sum first over the point masses that constitute extended body nr. 1 and then nr. 2 etc. (in your problem there are only two external bodies), then you can write:

R = 1/M Sum_{i} m_{i} r_{i} = 1/M [S1 + S2 + ...] (1.3)

where S1, S2 etc. are given by:

S1 = Sum_{i over body1} m_{i} r_{i} (1.4)

S2 = Sum_{i over body2} m_{i} r_{i} (1.5)

etc.

We can apply formula (1.1) to the summations in S1, S2, etc. to express these in terms of the center of masses of the extended masses. According to (1.1):

R1 = 1/M1 Sum_{i over body 1} m_{i} r_{i} (1.6)

R2 = 1/M2 Sum_{i over body 2} m_{i} r_{i} (1.7)

etc.

Using (1.4) and (1.5) you can rewrite (1.6) and (1.7) as:

R1 = 1/M1 S1

R2 = 1/M2 S2

So, the partial summations S1 and S2 are given by:

S1 = M1 R1

S2 = M2 R2

Inserting this in (1.3) gives:

R = 1/M [M1 R1 + M2 R2 + ...]

M is the total mass which is obviously the sum of the masses of the extended bodies and thus given by M1 +M2+...

2)

Let's put the semicircle on the upper half plane of the xy-plane. Introduce polar coordinates so that theta is the angle with the positive x-axis and a move in the counterclockwise direction corresponds to increasing theta. So, the positive y-axis is at theta = 90 degrees = pi/2 and the negative x-axis is at theta = 180 degrees = pi.

The semicircle covers the region from theta = 0 to theta = pi and r = 0 to r = R. By symmetry the center of mass will be on the y-axis, so we only have to find the y-coordinate of the center of mass. Let's introduce a surface mass density defined as the ...

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Intro to the Physics Waves

Some short-answer questions involving the basic vocabulary of string, sound, and water waves.

##### Variables in Science Experiments

How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.

##### Introduction to Nanotechnology/Nanomaterials

This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.

##### The Moon

Test your knowledge of moon phases and movement.

##### Basic Physics

This quiz will test your knowledge about basic Physics.