b) The general form for the energy of a simple harmonic oscillator is E=(1)/(2)mass(velocity)^2+(1)/(2)stiffness(displacement)^2 a) State and briefly explain three velocities in wave motion which are quite distinct although they are connected mathematically. 3mks Set up the energy equations for a simple pendulum and use the expression (dE)/(dt)=0 to derive the equation of motion in each case. 5mks c) Show that, in the Doppler effect, the change of frequency noted by a stationary observer 0 as a moving source S' passes him is given by Delta v=(2vcu)/((c^2)-u^(2)) where c=vlambda the signal velocity and u is the velocity of s. 4mks d) Define the term "Quality factor' as used in damped oscillators 1mk e) When the E string of a guitar (frequency 330 Hz) is plucked , the sound intensity decreases by a factor of 2 after 4 s Determine (i) the decay time T. (ii) the quality factor Q and (iii) the fractional energy loss per cycle. 2mks f) You are on a stationary ship collecting weather data from a mobile weather station in space. How will you tell if a mobile weather station is approaching or receding 2mks g) The sonar device on a fishing boat uses underwater sound to locate fish. Would you expect sonar to be a longitudinal or a transverse wave 2mks

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$b) The general form for the energy of a simple harmonic oscillator is E=(1)/(2)mass(velocity)^2+(1)/(2)stiffness(displacement)^2 a) State and briefly explain three velocities in wave motion which are quite distinct although they are connected mathematically. 3mks Set up the energy equations for a simple pendulum and use the expression (dE)/(dt)=0 to derive the equation of motion in each case. 5mks c) Show that, in the Doppler effect, the change of frequency noted by a stationary observer 0 as a moving source S' passes him is given by Delta v=(2vcu)/((c^2)-u^(2)) where c=vlambda the signal velocity and u is the velocity of s. 4mks d) Define the term "Quality factor' as used in damped oscillators 1mk e) When the E string of a guitar (frequency 330 Hz) is plucked , the sound intensity decreases by a factor of 2 after 4 s Determine (i) the decay time T. (ii) the quality factor Q and (iii) the fractional energy loss per cycle. 2mks f) You are on a stationary ship collecting weather data from a mobile weather station in space. How will you tell if a mobile weather station is approaching or receding 2mks g) The sonar device on a fishing boat uses underwater sound to locate fish. Would you expect sonar to be a longitudinal or a transverse wave 2mks$

b) The general form for the energy of a simple harmonic oscillator is E=(1)/(2)mass(velocity)^2+(1)/(2)stiffness(displacement)^2 a) State and briefly explain three velocities in wave motion which are quite distinct although they are connected mathematically. 3mks Set up the energy equations for a simple pendulum and use the expression (dE)/(dt)=0 to derive the equation of motion in each case. 5mks c) Show that, in the Doppler effect, the change of frequency noted by a stationary observer 0 as a moving source S' passes him is given by Delta v=(2vcu)/((c^2)-u^(2)) where c=vlambda the signal velocity and u is the velocity of s. 4mks d) Define the term "Quality factor' as used in damped oscillators 1mk e) When the E string of a guitar (frequency 330 Hz) is plucked , the sound intensity decreases by a factor of 2 after 4 s Determine (i) the decay time T. (ii) the quality factor Q and (iii) the fractional energy loss per cycle. 2mks f) You are on a stationary ship collecting weather data from a mobile weather station in space. How will you tell if a mobile weather station is approaching or receding 2mks g) The sonar device on a fishing boat uses underwater sound to locate fish. Would you expect sonar to be a longitudinal or a transverse wave 2mks

Answer 1

a) The three velocities in wave motion that are distinct but connected mathematically are: 1. Phase velocity: It is the velocity at which a wave propagates through a medium. It is given by the ratio of the angular frequency to the wave number. 2. Group velocity: It is the velocity at which the energy of a wave propagates through a medium. It is given by the negative derivative of the angular frequency with respect to the wave number. 3. Signal velocity: It is the velocity at which the wavefronts of a wave propagate through a medium. It is given by the product of the wave speed and the direction of propagation. b) The energy equations for a simple pendulum are: Potential energy: $E_p = mgh$ Kinetic energy: $E_k = \frac{1}{2}mv^2$ Using the expression $\frac{dE}{dt} = 0$, we can derive the equation of motion for a simple pendulum: $\frac{d}{dt}(E_p + E_k) = 0$ $\frac{d}{dt}(mgh + \frac{1}{2}mv^2) = 0$ $mg\frac{dh}{dt} + m\frac{dv}{dt} = 0$ $g\frac{dh}{dt} + v\frac{dv}{dt} = 0$ $g\sin(\theta) = -v\frac{dv}{dt}$ $g\sin(\theta) = -v\omega^2 \cos(\theta)$ $g\sin(\theta) = -\omega^2 l \sin(\theta)$ $\omega^2 = \frac{g}{l} \sin(\theta)$ $\omega = \sqrt{\frac{g}{l} \sin(\theta)}$ c) To show that the change of frequency noted by a stationary observer as a moving source passes him is given by $\Delta v = \frac{2cu}{(c^2 - u^2)}$, we can use the Doppler effect formula: $f' = \frac{f(c + u)}{c}$ where $f'$ is the observed frequency, $f$ is the emitted frequency, $c$ is the signal velocity, and $u$ is the velocity of the source. Rearranging the equation, we get: $f' - f = \frac{fu}{c}$ $\Delta v = \frac{fu}{c}$ $\Delta v = \frac{2cu}{(c^2 - u^2)}$ d) The term "Quality factor" in damped oscillators refers to the ratio of the natural frequency of oscillation to the bandwidth of the oscillator. It is a measure of the damping or energy loss in the oscillator. e) (i) The decay time $T$ can be determined using the formula: $T = \frac{1}{\text{decay constant}}$ (ii) The quality factor $Q$ can be calculated as: $Q = \frac{1}{\text{decay constant}}$ (iii) The fractional energy loss per cycle can be calculated as: $\text{Fractional energy loss} = \frac{1}{Q^2}$ f) To determine if a mobile weather station is approaching or receding, you can compare the frequency of the signals received from the weather station with the known frequency of the signals transmitted by the weather station. If the frequency of the received signals is higher than the transmitted frequency, the weather station is approaching. If the frequency of the received signals is lower than the transmitted frequency, the weather station is receding. g) Sonar devices on a fishing boat use underwater sound to locate fish. Sonar waves are longitudinal waves, as they involve the compression and rarefaction of the medium (water) in the direction of wave propagation.

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