Numerical on Doppler Effect

Shortcut Method: For a moving observer, the apparent frequency is given by:

$$f^{\prime }=\frac{v\pm v_0}{v\pm v_s}f$$

where:

• v is the speed of sound in air
• v0 is the speed of the observer
• vs is the speed of the source (e.g. police car)
• f is the actual frequency of the source (e.g. siren)

In this numerical:

• v = 344 m/s (speed of sound in air)
• v0 = 0 m/s (stationary observer)
• vs = 30 m/s (police car moving towards the observer)
• f = 500 Hz (siren frequency)

Substituting these values, we get:

$$f^{\prime }=\frac{344+0}{344-30}500$$ $$f^{\prime }=\frac{344}{314}\times 500$$ $$f^{\prime }=544.62\text{ Hz}$$

So the apparent frequency heard by the person is 544.62 Hz.

Numerical on Polarization of Light

Shortcut Method:

The intensity of the light transmitted by a polarizer is given by:

$$I=I_0{\cos }^2\theta$$

where:

• I is the intensity of the transmitted light
• I0 is the intensity of the incident light
• θ is the angle between the polarization of the incident light and the optical axis of the polarizer

In this numerical:

• I = 0.5I0 (intensity of the transmitted light)
• I0 is the intensity of the incident light
• θ = 30 degrees (angle of incidence)

Substituting these values, we get:

$$0.5I_0={\cos }^{2}30\text{degree}$$ $$\cos 30\text{degree}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$$ $$\theta =\arccos \left(\frac{\sqrt{2}}{2}\right)$$ $$\theta =45\text{degree}$$

So the angle of polarization of the incident light is 45 degrees to the optical axis of the polarizer.

Numerical on Electromagnetic Spectrum

Shortcut Method:

The frequency of a wave given by:

$$f=\frac{v}{\lambda }$$

where:

• f is the frequency of the wave
• v is the speed of the wave
• λ is the wavelength of the wave

In this numerical:

• v = 3 * 10^8 meters per second (speed of light)
• λ = 300 meters (wavelength of the radio wave)

Substituting these values, we get:

$$f=\frac{3\times {10}^8\text{ m/s}}{300\text{ m}}$$ $$f=1\times {10}^6\text{ Hz}$$

So the frequency of the radio wave is 1 MHz (1 * 10^6 Hz).