The angular momentum of a particle about a fixed axis is . As the particle draws nearer the fixed axis, both  and  change. However, the product  remains constant. If you imagine a triangle connecting the three points, the product  represents the  "of closest approach", labeled "" in the diagram.

Center of mass can be found by spinning an object. It will naturally spin around its center of mass, due to the concept of even distribution of mass in relation to the center of mass. Shape and mass are important factors in this property, but the most improtant factor is the mass distribution.

Changing the position of the fulcrum by moving it to the left means the center of mass will be to the right of the new position. Therefore, the scale will tip right. Adding more mass to the left end will rebalance the scale. None of the other options make sense. Adding more mass to the new fulcrum position will not change the balance of the scale because that mass is a negligible distance from the new fulcrum position and does nothing to change the masses on either side.

This question asks us to find the center of mass for this system. We know that the center of mass resides a distance  from the first mass such that:

In this case:

Plug in known values and solve.

To find the center of mass, we have to take the weighted average of the x coordinates and the y coordinates.

 Measures:                                        Measures

First, we take the weighted measurement of the x-axis:

We can see that the result of the x-axis contribution is equal to .

Now, let's look at the y-axis contribution:

This equals to 

Now that we have the x and y components, we take the root of squares to get the final answer:

This will give us 

Since it is experiencing a constant torque and constant angular acceleration, the angular displacement can be calculated using:

The angular acceleration is easily calculated using the angular velocity and the time:

Using this value, we can find the angular displacement:

Convert the angular displacement to revolutions by diving by :

We can find the angular acceleration using the rotaional motion equivalent of Newton's second law. In rotational motion, torque is the product of moment of inertia and angular acceleration: 

The moment of inertia for a circular disk is:

The tourque is the product of force and distance (in this case, the radius):

We can plug these into our first equation:

Simplify and rearrange to derive an equation for angular acceleration:

Use our given values to solve:

Torque is given by,

Since all of the forces are equal in magnitude, the magnitude of the torque is then influenced by the radius r and the angle theta between the radius and the force.

For ,

For ,

For ,

For ,

Combining this information yields the relationship,

The net torque on the beam is given by addition of the torques caused by the weight of the man and the weight of the beam itself, each at its respective distance from the end of the beam:

Let's assign the direction of positive torque in the direction of the torques of the man's and the beam's weights, noting that they will add together since they both point in the same direction.

We can further simplify by combining like terms:

Using the given numerical values,

A torque analysis is appropriate in this situation due to the inclusion of distances from a given pivot point. Generally,

This is a static situation. There are two torques about the pivot caused by the weights of two children. We will note that these weights cause torques in opposite directions about the pivot, such that

Consequently,

Or more simply,

Solving for ,