Speed of Light

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This is a phrase that often appears in student responses in Physics examinations. Let's look at various versions of it and see if they are correct. We will start from the most basic.

  1. Light is constant. This is incorrect as the property of light has not been identified.
  2. The speed of light is always constant. This is incorrect. The speed of light in water is less than the speed of light in a vacuum.
  3. The speed of light in a vacuum is a constant value. This is not fully correct as the frame of reference has not been included.
  4. The speed of light in a vacuum is the same value in any inertial frame of reference. Correct. This was one of Einstein's postulates in 1905.
  5. The speed of light in a vacuum is the same value in any frame of reference. Incorrect. In accelerating (non-inertial) frames of reference the speed of light in a vacuum is not always c.

Here are some other misconceptions about the speed of light.

  1. The speed of light in a vacuum is given the symbol c because c is the first letter of constant. Incorrect. The c comes from the Latin word celeritas for speed.

  2. The Michelson and Morley experiment showed that the speed of light was always constant. Incorrect. There is no mention of the constancy of the speed of light in their paper describing the experiment. Einstein proposed this in 1905. M and M were not able to detect the motion of the Earth relative to the aether (this is a null result)

  3. Light waves are different to sound waves. The speed of sound in air at 20 degrees C is 343m/s. Imagine that you move towards a source of sound at 20 m/s. What is the speed of the sound waves relative to you? [363 m/s]
  4. The Michelson-Morley experiment found that the light waves in their interferometer always arrived in phase. Incorrect. The rays of light produced an interference pattern after travelling on perpendicular paths. This pattern did not change when the apparatus was rotated through 90 degrees. This is called no fringe shift.
  5. The first determination of the speed of light was made by the Dutch physicist Christian Huygens in 1677 using astronomical measurements made by Olaus Roemer. The value obtained was 2.3x108 m/s. The French physicist Hippolyte Fizeau using an Earth based method determined the speed of light as 3.1x108 m/s in 1849.
“To Tom...for always and in all ways”
— Elizabeth, Catherine and Stephen

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Damping of Forced Vibrations

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In the May 2015 TZ1 IB Physics Paper 1 SL and HL there was a question (13SL,10HL) on the effect of damping on an oscillating system experiencing a driving force of variable frequency. The IB Examination Report states that "as damping is increased and friction is introduced into the system so the time per oscillation will increase, answer is B". This may not be intuitively obvious so a mathematical route to the solution may be helpful.

Let the frequency of vibration of the system when no damping occurs (usually just called the natural frequency) be F. The frequency of the driving force that acts on the system is f. The amplitude of the forced oscillations (this is the greatest displacement of the system from its equilibrium position) caused by the action of the driving force is A. Damping means that a resistive force is acting on the system as it vibrates causing its energy to decrease.

  1. Driving force with no damping. When damping is not present a huge vibration known as resonance ocurs when the frequency of the disturbing force equals the natural frequency of vibration. As the amplitude is huge we can predict that the amplitude of oscillation must be inversely proportional to the difference in the frequencies. When f - F approaches zero A approaches infinity. A mathematical solution of the differential equation describing the system without damping gives A = 1/(f2-F2). How can a catastrophic vibration be avoided when f = F? Including a damping term in the equations removes some energy from the system allowing it to be unavailable to participate in a large vibration.

  2. Driving force with damping. When a damping term is present the differential equation describing the system contains an additional term 2kv, where k is the damping coefficient and v is the velocity of the mass. The solution for A with damping is A = 1/√((f2-F2)2+ 4k2f2 ). Notice the additional term in the denominator. When f = F we have A = 1/(4k2f2) giving a finite amount of vibration when f = F. Catastrophe is avoided! Let us examine the equation

    A = 1/√( (f2-F2)2+ 4k2f2 )

If we plot A versus f we obtain the graph shown on the 2015 examination paper. By differentiating A with respect to f we find that A has a maximum value of A0 when f has the value f0

f0 = √( F2-2k2 )

A0 =1/( 2k √(F2-k2) )

To answer the question, as k increases A0 decreases and f0 decreases so alternative B is the correct answer.

 

 

Velocity-Time Graph

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Question 16 in the 2016 HSC Mathematics examination involved calculating the distance travelled by a particle given its velocity as a function of time. The velocity during the first second of the motion was negative. This trapped many students. To find the total distance travelled we must add up the absolute values of all of the areas under the v-t graph. Be careful! If you have done it correctly the answer is 10-4ln2. If you obtain 14-12ln2 this is the change in displacement or the final position of the particle measured from the starting point; this is the length of the arrow from the starting point to the finishing point and is not the distance travelled which is the total amount of ground covered.