We can use the expression for conservation of energy to solve this problem:

Given the problem statement, we know that the ball has no velocity at the initial and final states, so we can remove kinetic energy from the equation:

The initial potential energy is stored in the springs and the final potential energy is gravitational, so we can write:

We multiply by 25 because there are 25 springs.

Rearranging for mass, we get:

We know all of these values, allowing us to solve:

Initially the car is at rest at the bottom of a hill, this velocity and height are zero.

Converting to

Plugging in values:

We can use the expression for conservation of energy:

Since the rod is both initially and finally at rest, we can removed both kinetic energies. Also, if we assume point p is at a height of 0, we can removed initial potential energy, leaving us with:

Plugging in the expression for potential energy and expanding for both masses:

Since the rod is vertical, we know that mass A is half a rod's length above our reference height, and mass B is half a rod's length below it. Thus we get:

Factoring to clean up our expression:

We know all of our variables, so time to plug and chug:

We can use the expression for conservation of energy:

Since the rod is both initially and finally at rest, we can removed both kinetic energies. Also, if we assume point p is at a height of 0, we can removed initial potential energy, leaving us with:

Plugging in the expression for potential energy and expanding for both masses:

Since the rod is vertical, we know that mass A is half a rod's length above our reference height, and mass B is half a rod's length below it. Thus we get:

Factoring to clean up our expression:

We know all of our variables, so time to plug and chug:

Recall that work can be defined as:

Here, is the magnitude of the force vector, is the magnitude of the displacement vector, and is the angle between the directions of the force and displacement vectors. In the case of circular motion, the force vector is normal to the circle since it points inward, and the displacement vector is tangent to the circle. This means that the angle between the force vector and displacement is . Since , work done by the centripetal force on an object moving in a circle is always .

Let's use the midpoint as our point of reference. With that said, is from the center, and is from the center. However, each of these are on different sides of the midpoint. When we set up our center of gravity equation, we must determine a (+) and (-) side in order to denote under which side our mobile string will fall. It may be helpful to look at the mobile as a number line, with the left being negative, and the right side positive, but you can really use which ever is more comfortable.

Our main equation is this:

Where is our weight in newtons and is our distance in meters. Typically, we would want to convert everything to SI units, so let's go ahead and do that (ex: )

Now let's plug in our numbers, remember about our negative/positive sides!

This should give us

Now, because this number is negative, we know it's to the left of our midpoint (if you chose to set up your -/+ sides opposite of how we did it, your answer will be positive). Regardless of the outcome, this measurement is meant to be taken from object A to the midpoint. Well, do any of our answers have either (+) or (-) ? No. But if we read them carefully, we can determine that the answer is from the left of the dowel by using simple subtraction.

One side is

Think about it logically too (use your pencil). The heavier end of a pencil usually has a bulky eraser on it (just like the bulkier object on the left). Try holding the pencil at the tip's end, and then gradually try balancing it in the same manner as you move closer to the eraser.

ALSO NOTE: the question states the weight of the dowel as well. But look we didn't even need it. Sometimes the AP exam will give you some aspects in the question to distract you.

Using

Determining initial kinetic energy:

Combining equations

Converting to and plugging in values:

Determining final kinetic energy:

Combining equations

Converting to and plugging in values:

Plugging in values:

Work is defined as:

Where is work, is force experienced by the object/person, and is the distance the object/person moves as a result of the force. Recall that force is calculated as follows.

Where is mass and is acceleration; therefore, work depends on the mass and acceleration (force), and distance.

In general,

In order to calculate the work done by a force, we need to find the angle between the force and the displacement through which the object moves. Since the displacement is directed down the incline, then this means the friction force acts up the incline. This means . We must remember the frictional force is represented as

Where is the normal force and is the coefficient of kinetic friction. On an inclined plane, we can show that . Therefore, we can finally write

Plugging in everything, and noting the magnitude is the absolute value of a quantity, we can plug in for our answer:

We can assume that the dog must carry his entire weight up the tree, and therefore a force is exerted. Using the equation

we can use the evidence provided by the problem to solve for distance.