We are asked to determine what orientation of the rod will result in masses A and B having the same gravitational potential energy. Therefore, we first need the equation for gravitational potential energy:

We can then write this formula for each mass:

Where the subscripts denote each mass. Now we can set the two equations equal to each other (since that is the point of the question):

Canceling out g:

We are given both masses in the problem statement, so we need to determine the height of each mass. Let's think about this situation practically; Since mass A is greater than mass B, mass B will need to be higher up for the two gravitational potential energies to be the same. Therefore, we can determine that the rod will be rotated counter clockwise compared to the original figure, and  will be positive. With this in mind, we can create equations for the height of each mass. Let's start with mass A:

We have heigh h at which the mass is at when the rod is horizontal. As we just discussed, the mass will be slightly lower than this. How much lower? We can use the angle c and length of the rod to determine a distance d, which is how far the mass is below horizontal (imagine a right triangle between the mass, point p, and the horizontal position of mass A):

This is simply the sine function for right triangles. We can take this distance and subtract it from h to get the height of mass A:

We can do the same for mass B, except we will be adding the distance d since it will be above horizontal:

Substituting these into our reduced equation above we get:

Expanding our terms:

Getting height and angles on same sides:

Factoring each side of the equation:

Dividing so we get  alone on one side:

Taking the inverse sine of both sides:

Finally, we have arrived to our final equation. We know every value on the right side of the equation. However, lets do a quick check and make sure our units will work out (we want a unitless value inside the parenthesis).

Now it's time to plug and chug:

Using conservation of energy:

Solving for :

Plugging in values:

Using:

The escape velocity is the velocity necessary to have a kinetic energy that can overcome the gravitational potential. That is, the kinetic energy plus the "big G" gravitational potential energy need to add to zero or a positive number.

Solving for

Plugging in values:

The energy before must be equal to the energy when the object is in motion.

Determine the mass of the object.

Plug into the conservation of energy statement above. 

The final velocity of the object will be .

We must first convert to the proper units.

Now use the equation for kinetic energy.

Plug in and solve for the kinetic energy.

John Elway could throw a football with a kinetic energy of .

For this question we will use conservation of gravitational potential energy.

Plug in and solve for .

The man will gain  of gravitational potential energy by walking to the top of the building.

Use the equation for weight to determine the objects mass.

Plug in and solve for the objects mass.

Use the equation for the potential energy due to gravity.

Plug in and solve for the height.

The bucket can be lifted to a height of .

Using:

Combining the radius and the distance to the surface, converting to and plugging in values:

First, determine the final gravitation potential energy:

Using:

Combining the radius and the distance to the surface, converting to and plugging in values:

Then, determine the initial gravitational potential energy:

Using:

Combining the radius and the distance to the surface, converting to and plugging in values:

There are two forces acting on the coaster at the top of the loop: gravity and centrifugal. It is important to note in this problem the difference between centripetal and centrifugal forces. Centripetal force the is actual force applied to the coaster (points inward) and the centrifugal force is that apparent force felt by the coaster as it travels in a loop (points outward). Just remember that these forces are equal, but in opposite directions. Therefore, we will be using the centrifugal force to calculate the g's felt by the coaster. To do this, we will begin with the expression for conservation of energy:

If we assume that the bottom of the loop has a height of zero, we can eliminate initial potential energy. Also, the only work is done by friction, and it is work done on the surrounds, so it will be negative. Therefore, we get:

Plugging in expressions for each of these, we get:

Rearranging for final velocity, we get:

Where d is half the circumference of the loop:

And the final height is twice the radius:

Plugging these in, we get:

Plugging in our values, we get:

Now we can use this to calculate the centrifugal force:

This is in the opposite direction of the other force acting on the system, gravity. Therefore, we get:

Plugging in expressions we get:

Rearranging for net acceleration, we get:

Then calculating g's we get: