A 2.50-kg Pendulum is attatched to a string 11.4m stringis pulled aside until it is.386m above its lowest position and released. The pendulum is designed to emit sound waves at a frequency of 440 Hz; However, as it swings toward and away from an observer, the frequency appears to vary slightly. What is this Phenomenon called? What would be the apparent frequency of the sound from the pendulum as it swings at its maximum speed toward an observer? Assume the speed of sound is 345 m/s.

The frequency change is called the Doppler shift. Calculate maximum pendulum speed V, which occurs at the bottom of the swing's motion.

Conservation of Energy tells you that

(1/2)V^2 = g * 0.386 m

Solve for V. Then use the Doppler formula for the higher-pitched frequency at maximum velocity V.

The shift is approximately
delta f = fo * (V/345 m/s)
Add that to fo = 440 Hz for the frequency that is heard by a stationary listener

http://en.wikipedia.org/wiki/Christian_Doppler

The phenomenon you're referring to is known as the Doppler effect. It describes the change in frequency of a wave as an observer moves relative to the source of the wave. In this case, as the pendulum swings toward and away from the observer, the frequency of the sound waves emitted by the pendulum appears to vary slightly.

To calculate the apparent frequency of the sound from the pendulum as it swings at its maximum speed toward an observer, we need to consider the velocity of the pendulum, the velocity of sound, and the original frequency of the pendulum.

First, let's find the velocity of the pendulum at its lowest position using conservation of energy. The potential energy when the pendulum is pulled aside is converted to kinetic energy when it reaches its lowest position. The formula for potential energy is given by:

PE = mgh

where m is the mass of the pendulum (2.50 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height above the lowest position (0.386 m). Plugging in the values, we have:

PE = (2.50 kg)(9.8 m/s^2)(0.386 m)
= 9.56 J

Since the potential energy is converted entirely to kinetic energy, we can equate it to the kinetic energy at the lowest position:

KE = (1/2)mv^2

where v is the velocity of the pendulum at the lowest position. Plugging in the values, we have:

9.56 J = (0.5)(2.50 kg)v^2
19.12 J = (2.50 kg)v^2
7.648 = v^2
v ≈ 2.77 m/s

Now, let's use the Doppler effect formula to calculate the apparent frequency of the sound from the pendulum as it swings at its maximum speed toward an observer:

f' = f * (v_sound / (v_sound ± v_observer))

where f is the original frequency of the pendulum (440 Hz), v_sound is the speed of sound (345 m/s), and v_observer is the velocity of the observer. Since the pendulum is swinging towards the observer at its maximum speed, the observer's velocity is equal to the velocity of the pendulum (v = 2.77 m/s). Plugging in the values, we have:

f' = 440 Hz * (345 m/s / (345 m/s + 2.77 m/s))
= 440 Hz * (345 m/s / 347.77 m/s)
≈ 437.89 Hz

Therefore, the apparent frequency of the sound from the pendulum as it swings at its maximum speed toward an observer is approximately 437.89 Hz.