Displacement is the change in distance from start to finish

Where  is her position on the circular track. Since Laura starts and ends at the same place, the displacement is 0. It doesn't matter what we label her start/end position as, it's her initial position compared to her final.

Inertia is the ability for an object to resist motion. In this case, inertia is determined by how far away the mass (weights) are distributed from the dumbbell's COM (center of handle). While there is no definitive equation for the "inertia of a dumbbell", we can conceptualize this. All inertia equations deal with a mass and a radius. The larger either of these numbers are, the larger the inertia! We know that the larger the inertia, the more resistant that object is to change in motion (or the more awkward it would be to pick up that object). With that said, it would be harder for Steve to pick up the new weight. Not because of the weight itself, but because of how that same weight is distributed around that object. Think about it this way; a piece of paper weighs roughly .16 oz. if I gave you a 1 cm cube that weighed .16 oz, which would be easier to pick up and balance on your head? I would most likely assume that you would say the cube. This is because the weight of the cube is more closely positioned near the COM, rather than the thin sheet of paper with it's weight evenly spread throughout 8.5" x 11".

The bucket would proceed in the same direction as the tangential velocity at that given time. As seen in the figure below, each position of the bucket around the circle will give rise to a new tangential velocity vector. If the bucket is going counter clockwise and is released at the "3 o'clock" position, the  will be in the upward direction.

We know from the question that

Frequency is the inverse of period:

The velocity of a wave is its wavelength multiplied by its frequency:

The doppler effect follows this formula:

In this equation,  is the new frequency you will hear,  is the speed of sound,  is the velocity of the moving sound-emitting thing, and  is the initial frequency of the sound.

Plugging the given values in, we can describe the initial situation as:

Note that the velocity is negative  because the car is driving towards you.

Therefore,

When the police car is driving away, the situation is described with a positive velocity:

Therefore,

Two waves will emit a beat with a fequency equal to the difference in frequency of the two waves. In this case the beat frequency is:

    (beats per second)

Convert to beats per minute:

The speed of sound in air is constant, assuming that the temperature of the air is constant. When the length of the string is shortened, by the principles of standing waves, this creates a higher frequencies. Assuming that the two instruments use the same strings is equivalent to stating that the two instruments have strings of equal linear mass density. This situation represents a standing wave, thus we can relate the following equation for the first harmonic:

Where,  is the length of the string and  is the wavelength. Then we can use the following equation to relate wavelength and speed (which is known) to frequency:

Since the velocity of sound in a fixed medium is constant, we see that a shorter length,  corresponds to a shorter wavelength, . Thus when  decreases, frequency,  must increase, to keep velocity constant. 

Intensity is related to radius by the inverse square law: 

This equation is derived from the concept that the energy from the sound waves is conserved and spread out over an area, producing the  term. Applying this concept, when the radius doubles, the intensity decreases by a factor of 4. The correct answer is . 

A stopped pipe can be modeled with the following equation: 

Rearrange the equation to solve for L, then plug in given values and solve.

Frequency of beats is determined by the absolute value of the difference between two different frequencies. Thus, the beats frequency is 2Hz. Note that beat frequency is always a positive number.