Assume a constant acceleration we may use a kinematic formula:

 is the acceleration due to gravity, which is 

, the initial velocity, is zero

 is the unknown

 is the ground, which will be a height of

Plug in values:

Solve for

Gravitational potential energy is calculated using the following equation.

Where  is the potential energy,  is the gravitational constant,  is the mass of one object,  is the mass of the other object, and  is the distance between the two objects. Decreasing  and increasing mass of the object by the same factor will have a similar effect on the potential energy (both will increase potential energy by the same factor); therefore, changing both of these variables by same factor will have a similar effect. Recall that the force due to gravity is 

When calculating ,  will have the biggest effect. Don't confuse equation for potential energy with equation for force due to gravity.

Potential energy is calculated using the following equation.

Where  is the potential energy,  is the gravitational constant,  is the mass of one object,  is the mass of the other object, and  is the distance between the two objects. The potential energy depends on the mass of the objects (in this case the person and the planet); therefore, the more massive planet will produce the higher potential energy.

The mass is the same regardless of the situation. Mass measures the amount of substance/elements/molecules inside a person or object's body. The composition of the person’s body won’t change (regardless of the mass of the planet); therefore, the mass will stay the same. Recall that the weight of the person, however, does change based on the planet the person is standing on. This is because the weight depends on the gravitational acceleration, which depends on the mass of the planet.

The potential energy of a person or an object on earth is calculated using the following formula.

Where  is potential energy,  is mass,  is acceleration due to gravity, and  is the height from surface of earth (sea level). The question states that the person is standing on the surface of earth; therefore, . This means that the person has no potential energy.

Recall that kinetic energy, on the other hand, is the energy associated with motion. It depends on the velocity of an object or person. A person on the surface of earth will have kinetic energy if the person is moving.

Definition of work:

Combine equations:

Substitute variables, then plug in and solve for the change in heat.

At the top of the loop, the centripetal acceleration will need to be at least the acceleration due to gravity:

Solve for

The radius of the circle will be half the height:

Use conservation of energy:

Solve for :

Combine equations:

Plug in values:

In order to solve this problem, we will be using the expression for gravitational potential energy:

Writing this out for each mass, we get:

We are looking for the difference between potential energy, so we can write:

We know both masses and g, so we just need to determine the height of each to solve this problem. The statement states that , and according to the law of alternate angles, we know that the angle between mass B and horizontal is also . With this and the fact that point p is at the midpoint between masses, we know that the vertical distance between mass A and horizontal is the same as mass B and horizontal, just in opposite directions. Imagine mass A, the horizontal, and point p create a right triangle. We can then write:

Where d is the vertical distance of each mass from the horizontal. From the statement, we know that mass A is below the horizontal, and mass B is above; Thus, we can write:

Substituting these into our equation for difference in potential energy, we get:

We now have an equation we can plug and chug (you can definitely keep reducing it, but it's more work than needed. See the final equation at the end of this solution to see what you would have if interested).

Do a quick unit check to make sure we end up with joules.

Now to plug and chug:

We will only use the expression for gravitational potential energy to solve this problem:

Written out for each mass:

We are asked to find at which point does mass A have twice as much gravitational potential energy as mass B:

Now we need to determine the heights of each mass which are the height given in the problem +/- half the length of the rod:

Plugging these into our expression, we get:

We know all the values in this expression, so we just need to isolate h:

Expanding both sides:

Rearranging to get h on the same side:

Factoring out h and rearranging:

This is our final equation, time to plug and chug:

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: