This is a centripetal force problem. In this case the tension on the rope is the centripetal force that keeps the ball moving on a circle.

If we want for the rope not to break, then the tension should never exceed .

Now we just solve for velocity:

The correct answer is moment of inertia. For linear equations, mass is what resists force and causes lower linear accelerations. Similarly, in rotational equations, moment of inertia resists torque and causes lower angular accelerations. 

Just as force causes linear acceleration, torque causes angular acceleration. This can be seen most in the linear-rotational comparison of Newton's second law:

Linear (tangential) velocity,  is given by the following equation: 

Here,  is the angular velocity in radians per second and  is the radius in meters.

Solve.

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 :