Grade 11 : Physics - Oscillations (Set 2)

Explanation: In Simple Harmonic Motion, the restoring force is directly proportional to the displacement from the mean position and acts in the opposite direction: $F = -kx$, so $F \propto -x$.

Explanation: The time period of a simple pendulum for small angular displacements is given by $T = 2\pi \sqrt{\frac{l}{g}}$.

Explanation: The angular frequency for a spring-mass system is $\omega = \sqrt{\frac{k}{m}}$. Since $T = \frac{2\pi}{\omega}$, the time period is $T = 2\pi \sqrt{\frac{m}{k}}$.

Explanation: The speed in SHM is $v = \omega \sqrt{A^2 - x^2}$. Maximum speed occurs at the mean position ($x = 0$), giving $v_{max} = \omega A$.

Explanation: Acceleration in SHM is given by $a = -\omega^2 x$. The maximum magnitude occurs at the extreme position ($x = A$), yielding $a_{max} = \omega^2 A$.

Explanation: At the extreme positions, the velocity of the particle momentarily becomes zero ($v = 0$). Hence, its kinetic energy $K = \frac{1}{2} m v^2 = 0$.

Explanation: Total energy is the sum of kinetic and potential energy: $E = K_{max} = \frac{1}{2} m v_{max}^2 = \frac{1}{2} m \omega^2 A^2$.

Explanation: Frequency $f$ is the reciprocal of the time period $T$: $f = \frac{1}{T} = \frac{1}{0.05} = 20\text{ Hz}$.

Explanation: Angular frequency is calculated as $\omega = 2\pi f = 2\pi (5) = 10\pi\text{ rad/s}$.

Explanation: If displacement is $x = A \sin(\omega t)$, velocity is $v = A \omega \cos(\omega t) = A \omega \sin(\omega t + \frac{\pi}{2})$. The phase difference is $\frac{\pi}{2}\text{ rad}$.

Explanation: Since $a = -\omega^2 x = \omega^2 A \sin(\omega t + \pi)$, acceleration leads displacement by $\pi\text{ rad}$ ($180^\circ$).

Explanation: Amplitude is defined as the maximum displacement of the oscillating particle from its mean or equilibrium position.

Explanation: Speed is maximum at the mean position ($x = 0$), so kinetic energy $K = \frac{1}{2} m v^2$ attains its maximum value at $x = 0$.

Explanation: Time period $T \propto \sqrt{l}$. If length becomes $4l$, new time period $T' = 2\pi \sqrt{\frac{4l}{g}} = 2 T$, so it doubles.

Explanation: Time period $T = 2\pi \sqrt{\frac{m}{k}}$, so $T \propto \sqrt{m}$. Multiplying mass by 4 doubles the period: $\sqrt{4} = 2$.

Explanation: Substituting $t = 0$ into $x(t) = 4 \sin(10 \times 0) = 4 \sin(0) = 0m$.

Explanation: Dissipative forces (like friction or air resistance) reduce energy, causing amplitude to decrease exponentially over time.

Explanation: Resonance occurs when the driving frequency equals the system's natural frequency, maximizing energy transfer and amplitude.

Explanation: A second's pendulum takes 1 second to swing from one extreme to another, so its full period (there and back) is 2 seconds.

Explanation: Potential energy $U = \frac{1}{2} k x^2$ equals total energy half value $\frac{1}{2} (\frac{1}{2} k A^2) \implies x^2 = \frac{A^2}{2} \implies x = \frac{A}{\sqrt{2}}$.