Solve each question using grass method.
1. Calculate the period T, in seconds (s), for the following cyclical events:
a. 3 classes every 210 minutes
b. 5 swings of a pendulum in 6 s
2. Calculate the frequency f, in hertz (Hz), for the following cyclical events:
a. 210 oscillations in 5.0 s
b. 25 revolutions of a turntable in 0.5 minute
3. As a train pulls out of the station going 45 m/s it blasts its horn, what is the frequency heard by the train if the passengers still at the station are hearing 373 Hz?
1.
a. To find the period T, we need to divide the total time by the number of events. In this case, we have 3 classes every 210 minutes.
T = 210 minutes / 3 = 70 minutes.
Since we want the period in seconds, we convert 70 minutes to seconds by multiplying by 60:
T = 70 minutes * 60 seconds/minute = 4200 seconds.
b. To find the period T, we need to divide the total time by the number of events. In this case, we have 5 swings of a pendulum in 6 seconds.
T = 6 seconds / 5 = 1.2 seconds.
2.
a. To find the frequency f, we need to divide the number of events by the total time. In this case, we have 210 oscillations in 5.0 seconds.
f = 210 oscillations / 5.0 seconds = 42 Hz.
b. To find the frequency f, we need to divide the number of events by the total time. In this case, we have 25 revolutions of a turntable in 0.5 minutes.
f = 25 revolutions / 0.5 minutes = 50 Hz.
3. The frequency heard by the train is affected by the Doppler effect. The formula for calculating the frequency heard by the train is:
f' = f * (v + v0) / (v + vs)
where f' is the frequency heard by the train, f is the frequency heard by the passengers at the station, v is the speed of sound, v0 is the speed of the train, and vs is the speed of the source (in this case, the speed of sound is not given, but it is approximately 343 m/s).
Plugging in the given values:
f' = 373 Hz * (343 m/s + 45 m/s) / (343 m/s + 0 m/s)
f' = 373 Hz * (388 m/s) / (343 m/s)
f' ≈ 420.96 Hz
So, the frequency heard by the train is approximately 420.96 Hz.
What is the formula used for all three questions?
The formula used to solve the questions in all three cases is:
Frequency (f) = 1 / Period (T)
This formula relates the frequency of a cyclical event to its period. The reciprocal of the period gives the frequency, and vice versa.
What formula was used for question 1?
For question 1, the formula used was:
T = Total time / Number of events
This formula calculates the period T (in seconds) by dividing the total time by the number of events.
1. To calculate the period T, we use the formula T = 1 / f, where f is the frequency.
a. We know that 3 classes occur every 210 minutes. To find the period, we divide 210 minutes by 3: 210 minutes / 3 = 70 minutes. However, the period needs to be in seconds, so we convert 70 minutes to seconds by multiplying it by 60: 70 minutes * 60 seconds/minute = 4200 seconds. Therefore, the period T for this cyclical event is 4200 seconds.
b. We know that 5 swings of a pendulum occur in 6 seconds. To find the period, we divide 6 seconds by 5: 6 seconds / 5 = 1.2 seconds. Therefore, the period T for this cyclical event is 1.2 seconds.
2. To calculate the frequency f, we use the formula f = 1 / T, where T is the period.
a. We know that there are 210 oscillations in 5.0 seconds. To find the frequency, we divide 1 by 5.0 seconds/210 oscillations: 1 / (5.0 seconds / 210 oscillations) = 42 Hz. Therefore, the frequency f for this cyclical event is 42 Hz.
b. We know that there are 25 revolutions of a turntable in 0.5 minute. To find the frequency, we first need to convert 0.5 minute to seconds by multiplying it by 60: 0.5 minute * 60 seconds/minute = 30 seconds. Then, we divide 1 by 30 seconds/25 revolutions: 1 / (30 seconds / 25 revolutions) = 0.833 Hz. Therefore, the frequency f for this cyclical event is 0.833 Hz.
3. To calculate the frequency heard by the train, we use the formula f' = (v + v0) / (v - vs) * f, where f' is the frequency heard by the train, v is the velocity of sound (assumed to be constant), v0 is the velocity of the train, vs is the velocity of the source (in this case, the horn of the train), and f is the frequency heard by the passengers at the station.
In this case, we have:
v = velocity of sound = constant
v0 = velocity of the train = 45 m/s
vs = 0 m/s (stationary source at the station)
f = frequency heard by the passengers at the station = 373 Hz
Substituting the values into the formula:
f' = (v + v0) / (v - vs) * f = (v + 45 m/s) / (v - 0 m/s) * 373 Hz
Since we don't have the value for the velocity of sound (v), we cannot calculate the frequency heard by the train without that information.
To solve these questions using the grass method, we need to understand the formula for calculating the period and frequency of cyclical events.
The period (T) represents the time it takes to complete one full cycle of a cyclical event. It is measured in seconds (s). The formula to calculate the period is:
T = (Total time) / (Total cycles)
On the other hand, the frequency (f) represents the number of cycles that occur in one second (or Hertz). The formula to calculate frequency is:
f = (Total cycles) / (Total time)
Now, let's solve each question using the grass method.
Question 1a:
We are given that there are 3 classes every 210 minutes.
To find the period (T), we divide the total time (210 minutes) by the total cycles (3):
T = 210 minutes / 3
T = 70 minutes
Since the period needs to be in seconds, we convert it to seconds by multiplying by 60:
T = 70 minutes * 60 seconds/minute
T = 4200 seconds
So, the period for 3 classes every 210 minutes is 4200 seconds.
Question 1b:
We are given that there are 5 swings of a pendulum in 6 seconds.
To find the period (T), we divide the total time (6 seconds) by the total cycles (5):
T = 6 seconds / 5
T = 1.2 seconds
So, the period for 5 swings of a pendulum in 6 seconds is 1.2 seconds.
Question 2a:
We are given that there are 210 oscillations in 5.0 seconds.
To find the frequency (f), we divide the total cycles (210) by the total time (5.0 seconds):
f = 210 / 5.0
f = 42 Hz
So, the frequency for 210 oscillations in 5.0 seconds is 42 Hz.
Question 2b:
We are given that there are 25 revolutions of a turntable in 0.5 minutes.
First, we convert 0.5 minutes to seconds by multiplying by 60:
Total time = 0.5 minutes * 60 seconds/minute
Total time = 30 seconds
To find the frequency (f), we divide the total cycles (25) by the total time (30 seconds):
f = 25 / 30
f = 0.83 Hz
So, the frequency for 25 revolutions of a turntable in 0.5 minutes is 0.83 Hz.
Question 3:
We are given that the train is moving at a speed of 45 m/s and the passengers at the station hear a frequency of 373 Hz.
To find the frequency heard by the train, we use the formula for the Doppler effect:
f' = (v + vd)/(v + vs) * f
where
f' = frequency heard by the train
v = speed of sound in air (approximately 343 m/s)
vd = velocity of the train
vs = velocity of the source (stationary)
Given:
f = 373 Hz
v = 343 m/s
vd = 45 m/s
vs = 0 m/s
Substituting these values into the formula:
f' = (343 + 45) / (343 + 0) * 373
f' = 388/343 * 373
f' = 421.70 Hz
So, the frequency heard by the train is approximately 421.70 Hz.