Simple Harmonic Motion and Circular Motion

Simple harmonic motion is closely related to a uniform circular motion.

A real life simple harmonic motion.

Review Question 1

Answer the following questions as you watch the first YouTube video(Simple harmonic motion and uniform circular motion).

  1. How does the centre of the red cylinder and the green ball relate?
  2. At which part of the SHM(left) is the red cylinder moving the fastest and slowest?
  3. At which part of the circular motion(right) is the green ball moving vertically the fastest and slowest?

Review Question 2

Navigate to this site. This animation shows how a 4-stroke engine internal combustion works.

Answer the following questions:

piston

  1. The piston is moving with SHM. At which point is the piston the fastest?
  2. At which point is the piston moving the slowest?

piston2

Given that the maximum velocity of the piston is v,

  1. write down the expression of the velocity of the piston at the position shown in the diagram.
  2. write down the expression of the acceleration of the piston at the position shown in the diagram.
  3. Sketch a velocity-time graph of the SHM demonstrated by the piston. Draw two complete cycles and label the parts A, B and C on your graph.

 

Ultrasound Intensity Reflection and Attenuation

Intensity Reflection Coefficient

When a beam of ultrasound is directed into a body, a certain proportion of the initial intensity would be reflected at the boundary. The ratio of the reflected intensity to the initial intensity can be obtained using the formula

\begin{aligned}\frac{I_r}{I_0}=\frac{(Z_2-Z_1)^2}{(Z_2+Z_1)^2}\end{aligned}

ultrasound

Air-Tissue Boundary

Using the formula from above, we can calculate that the intensity reflection coefficient of an air-tissue boundary is more than 99%.

\begin{aligned}\frac{I_r}{I_0}=\frac{(1.63 - 0.0004)^2}{(1.63 + 0.0004)^2}=0.999\end{aligned}

Note that the squaring of both the numerator and the denominator means that it does not matter which medium is assigned as Z_1 and Z_2 .

This is the reason why impedance matching need to be done when ultrasound is performed on a patient.

Intensity Attenuation

When ultrasound travels across a tissue, its intensity attenuates. This is because the wave loses energy to the tissue as it travels. The intensity of the wave at any particular depth of the tissue is determined by the formula

I=I_0 e^{-mu x}

where x is the depth of the point at which the intensity is determined, $latex μ $ is the absorption coefficient of the material and I_0 is the original intensity before the ultrasound wave enters the particular medium.

ultrasound 2

To find the intensity of the ultrasound that is received at the edge of medium 1, we can employ the following sequence of calculations:

  1. I_0 is the initial intensity.
  2. When the wave reaches the boundary of medium 1 and medium 2, it would have been attenuated by the amount I=I_0 e^{-\mu x} where x  is the depth of medium 1.
  3. This wave is then reflected at the boundary, where \begin{aligned} I_2 = \frac{(Z_2-Z_1)^2}{Z_2+Z_1)^2}\end{aligned}.
  4. Lastly, when the wave reaches the edge of medium 1, its final intensity I_3=I_2 e^{-\mu x}

Summary

  1. When you are calculating the reflection of ultrasound at a boundary, use \frac{I_r}{I_0}=\frac{(Z_2-Z_1)^2}{(Z_2+Z_1)^2}
  2. When you are calculating the intensity attenuation over a distance, use I=I_0 e^{-\mu x}

Problems