A commuter train passes a passenger platform at a constant speed of 40.4 m/s. The train horn is sounded at its characteristic fr
1 answer:
(a) -83.6 Hz
Due to the Doppler effect, the frequency of the sound of the train horn appears shifted to the observer at rest, according to the formula:
where
f' is the apparent frequency
v = 343 m/s is the speed of sound
is the velocity of the source of the sound (positive if the source is moving away from the observer, negative if it is moving towards the observer)
f is the original frequency of the sound
Here we have
f = 350 Hz
When the train is approaching, we have
So the frequency heard by the observer on the platform is
When the train has passed the platform, we have
So the frequency heard by the observer on the platform is
Therefore the overall shift in frequency is
And the negative sign means the frequency has decreased.
(b) 0.865 m
The wavelength and the frequency of a wave are related by the equation
where
v is the speed of the wave
is the wavelength
f is the frequency
When the train is approaching the platform, we have
v = 343 m/s (speed of sound)
f = f' = 396.7 Hz (apparent frequency)
Therefore the wavelength detected by a person on the platform is
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(a) The electrons move horizontally from west to east, while the magnetic field is directed downward, toward the surface. We can determine the direction of the force on the electron by using the right-hand rule:
- index finger: velocity --> due east
- middle finger: magnetic field --> downward
- thumb: force --> due north
However, we have to take into account that the electron has negative charge, therefore we have to take the opposite direction: so, the magnetic force is directed southwards, and the electrons are deflected due south.
b) From the kinetic energy of the electrons, we can find their velocity by using
where K is the kinetic energy, m the electron mass and v their velocity. Re-arranging the formula, we find
The Lorentz force due to the magnetic field provides the centripetal force that deflects the electrons:
where
q is the electron charge
v is the speed
B is the magnetic field strength
m is the electron mass
r is the radius of the trajectory
By re-arranging the equation, we find the radius r:
And finally we can calculate the centripetal acceleration, given by:
- index finger: velocity --> due east
- middle finger: magnetic field --> downward
- thumb: force --> due north
However, we have to take into account that the electron has negative charge, therefore we have to take the opposite direction: so, the magnetic force is directed southwards, and the electrons are deflected due south.
b) From the kinetic energy of the electrons, we can find their velocity by using
where K is the kinetic energy, m the electron mass and v their velocity. Re-arranging the formula, we find
The Lorentz force due to the magnetic field provides the centripetal force that deflects the electrons:
where
q is the electron charge
v is the speed
B is the magnetic field strength
m is the electron mass
r is the radius of the trajectory
By re-arranging the equation, we find the radius r:
And finally we can calculate the centripetal acceleration, given by: