A distant galaxy is simultaneously rotating and receding from the earth. As the drawing shows, the galactic center is receding from the earth at a relative speed of uG = 1.90 106 m/s. Relative to the center, the tangential speed is vT = 3.00 105 m/s for locations A and B, which are equidistant from the center. When the frequencies of the light coming from regions A and B are measured on earth, they are not the same and each is different than the emitted frequency of 6.20 1014 Hz. Find the measured frequency for the light from each of the following.
To find the measured frequency for the light from each region A and B, we need to consider the effects of the Doppler shift due to the motion of the distant galaxy.
The Doppler shift occurs when there is relative motion between a source emitting waves and an observer. It causes a shift in the observed frequency of the waves.
The observed frequency, f_observed, can be calculated using the formula:
f_observed = (speed of light + observer's speed towards/away from the source) / (speed of light + source's speed towards/away from the observer) × f_emitted
In this case, the relative velocity between the galaxy and Earth is given as uG = 1.90 × 10^6 m/s.
For region A, the tangential speed, vT = 3.00 × 10^5 m/s. The direction of rotation does not affect the frequency shift, so we only consider the tangential speed.
To find the frequency shift for region A, we need to calculate the relative velocities between the observer (Earth) and the source (region A) in the formula.
The relative velocity between the observer and region A can be calculated as:
v_relative_A = v_observer - v_source
Since region A is rotating in the same direction as the receding motion of the galactic center, the velocities add up:
v_relative_A = uG + vT
Now we can plug in the values into the formula:
f_observed_A = (speed of light + v_relative_A) / (speed of light) × f_emitted
Using the given values:
speed of light = 3.00 × 10^8 m/s
f_emitted = 6.20 × 10^14 Hz
Substituting these values and calculating f_observed_A:
f_observed_A = (3.00 × 10^8 + (1.90 × 10^6 + 3.00 × 10^5)) / (3.00 × 10^8) × (6.20 × 10^14)
Now we can simplify and calculate f_observed_A.
Similarly, we can repeat these steps to find the measured frequency for light from region B. The only difference will be the relative velocity between the observer and region B:
v_relative_B = v_observer - v_source
Since region B is equidistant from the galactic center as region A, the relative velocity for region B will be the same:
v_relative_B = v_relative_A = uG + vT
Again, plug in the values and calculate f_observed_B.
Following these steps will allow us to find the measured frequency of light from regions A and B in the given scenario.