Periodic Motion and Mechanical Waves (4A) - Chem & Physics

1. Find the wavelength if a wave has speed $v=12\ \text{m/s}$ and frequency $f=3\ \text{Hz}$.

   $\lambda=4\ \text{m}$. Wavelength calculates as speed divided by frequency, applying the fundamental wave relation $v=f\lambda$ to given values.

2. Identify the phase difference in radians for two points separated by $\frac{\lambda}{2}$ on a sinusoidal wave.

   $\Delta\phi=\pi$. Phase difference scales with path length over wavelength times $2\pi$, yielding $\pi$ for half-wavelength separation in sinusoidal waves.

3. Find the new period of a mass-spring system if the spring constant changes from $k$ to $4k$ (same $m$).

   $T\rightarrow \frac{T}{2}$. Period inversely scales with square root of spring constant, halving when constant quadruples per $T\propto^1/\sqrt{k}$.

4. State the SHM relation between acceleration and displacement for a mass on a spring.

   $a=-\omega^2 x$. In SHM, acceleration is proportional to negative displacement, with $\omega^2$ as the constant from restoring force dynamics.

5. What physical property of the medium determines wave speed, and what property does not determine it?

   Speed set by medium; frequency set by source (not medium). Medium properties like density and elasticity dictate wave speed, while source determines frequency independently of the medium.

6. State the period of a mass-spring oscillator in terms of mass $m$ and spring constant $k$.

   $T=2\pi\sqrt{\frac{m}{k}}$. The period derives from solving the differential equation for a mass-spring system, depending on the square root of mass over spring constant.

7. Find the new period of a simple pendulum if its length changes from $L$ to $4L$ (same $g$).

   $T\rightarrow 2T$. Pendulum period depends on square root of length, doubling when length quadruples as $T\propto\sqrt{L}$ under constant gravity.

8. State the period of a small-angle simple pendulum in terms of length $L$ and gravity $g$.

   $T=2\pi\sqrt{\frac{L}{g}}$. For small angles, pendulum period approximates from torque balance, scaling with square root of length over gravitational acceleration.

9. What is the maximum speed of a mass in SHM in terms of $A$ and $\omega$?

   $v_{\max}=A\omega$. Maximum speed in SHM occurs at equilibrium, derived from velocity function or energy conservation as amplitude times angular frequency.

10. Find the period $T$ if a wave has frequency $f=8\ \text{Hz}$.

    $T=\frac{1}{8}\ \text{s}=0.125\ \text{s}$. Period is the reciprocal of frequency, converting cycles per second to time per cycle for oscillatory or wave motion.

11. Find the new period of a mass-spring system if the mass changes from $m$ to $4m$ (same $k$).

    $T\rightarrow 2T$. Period scales with square root of mass, so quadrupling mass doubles the period via $T\propto\sqrt{m}$ in mass-spring systems.

12. What is the maximum acceleration of a mass in SHM in terms of $A$ and $\omega$?

    $a_{\max}=A\omega^2$. Maximum acceleration in SHM happens at maximum displacement, equaling amplitude times angular frequency squared from the defining equation.

13. What condition must hold for a pendulum to be treated as simple harmonic motion?

    Small angles: $\sin\theta\approx\theta$ (in radians). The small-angle approximation linearizes the pendulum equation, enabling SHM by equating sine to the angle in radians.

14. Identify the phase difference in radians for two points separated by $\frac{\lambda}{4}$ on a sinusoidal wave.

    $\Delta\phi=\frac{\pi}{2}$. For sinusoidal waves, phase shifts by $2\pi$ per wavelength, so quarter-wavelength gives $\pi/2$ radians difference.

15. State the wave speed on an ideal string in terms of tension $T$ and linear density $\mu$.

    $v=\sqrt{\frac{T}{\mu}}$. String wave speed balances tension restoring force and inertial mass density, yielding the square root expression for transverse waves.

16. What is the definition of wavelength $\lambda$ for a traveling periodic wave?

    Distance between points in phase (e.g., crest to crest). Wavelength measures the spatial period of a wave, defined as the shortest distance between identical phase points in the cycle.

17. What is the spring potential energy as a function of displacement $x$ for an ideal spring?

    $U=\frac{1}{2}kx^2$. Spring potential stores elastic energy quadratically with displacement, following Hooke's law integration for conservative force.

18. What is the kinetic energy of a mass in SHM in terms of $m$ and instantaneous speed $v$?

    $K=\frac{1}{2}mv^2$. Kinetic energy in SHM represents motion energy, calculated classically as half mass times velocity squared at any instant.

19. What is the key distinction between transverse and longitudinal mechanical waves?

    Transverse: oscillation $\perp$ travel; longitudinal: $\parallel$ travel. Wave types differ by particle oscillation direction relative to propagation: perpendicular for transverse, parallel for longitudinal.

20. Find the frequency if a wave has speed $v=20\ \text{m/s}$ and wavelength $\lambda=5\ \text{m}$.

    $f=4\ \text{Hz}$. Frequency derives from speed over wavelength, using $f=v/\lambda$ for the provided wave parameters.

21. State the angular frequency of a mass-spring oscillator in terms of $m$ and $k$.

    $\omega=\sqrt{\frac{k}{m}}$. Angular frequency emerges from the SHM equation of motion, as the square root of spring constant over mass for oscillatory behavior.

22. What is the relationship among period $T$, frequency $f$, and angular frequency $\omega$?

    $f=\frac{1}{T}$ and $\omega=2\pi f=\frac{2\pi}{T}$. Frequency is the reciprocal of period, and angular frequency equals $2\pi$ times frequency, linking time-based oscillatory parameters.

23. State the total mechanical energy of an ideal mass-spring oscillator in terms of $k$ and amplitude $A$.

    $E=\frac{1}{2}kA^2$. Total energy conserves as maximum potential energy at amplitude extremes, expressed via spring constant and squared amplitude.

24. State the standard displacement function for simple harmonic motion using amplitude $A$ and phase $\phi$.

    $x(t)=A\cos(\omega t+\phi)$. This equation models oscillatory displacement as a cosine function, capturing amplitude, angular frequency, time, and initial phase.

25. State the speed of a wave in terms of frequency $f$ and wavelength $\lambda$.

    $v=f\lambda$. Wave speed equals frequency times wavelength, relating temporal and spatial periodicity for propagating waves.

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