Grade 11 : Physics - Oscillations (Set 1)

Explanation:Simple Harmonic Motion is defined by a restoring force that is directly proportional to displacement from the mean position and acts in the opposite direction ($F = -kx$). This leads to acceleration $a = -\omega^2 x$.

Explanation:The time period of a simple pendulum is given by:
$$T = 2\pi \sqrt{\frac{L}{g}}$$
Substituting $L = 1m$ and $g = \pi^2m/s^2$:
$$T = 2\pi \sqrt{\frac{1}{\pi^2}} = 2\pi \left(\frac{1}{\pi}\right) = 2\text{ s}$$

Explanation:Comparing $x(t) = 0.05 \cos(10\pi t + \pi/4)$ with standard equation $x(t) = A \cos(\omega t + \phi)$ gives:
$$\omega = 10\pi\text{ rad/s}$$

Explanation:Acceleration in SHM is given by $a = -\omega^2 x$. The magnitude $|a| = \omega^2 |x|$ is maximum when displacement $|x|$ is maximum, which occurs at the extreme positions ($x = \pm A$).

Explanation:If displacement is $x = A \sin(\omega t)$, velocity is $v = \frac{dx}{dt} = A \omega \cos(\omega t) = A \omega \sin\left(\omega t + \frac{\pi}{2}\right)$. Thus, velocity leads displacement by a phase angle of $\pi/2\text{ rad}$ ($90^\circ$).

Explanation:Angular frequency $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{80}{0.2}} = \sqrt{400} = 20\text{ rad/s}$.
Frequency $f = \frac{\omega}{2\pi} = \frac{20}{2\pi} = \frac{10}{\pi}\text{ Hz}$.

Explanation:Time period is $T = 2\pi \sqrt{\frac{m}{k}}$. When mass is quadrupled ($4m$), the new time period is $T' = 2\pi \sqrt{\frac{4m}{k}} = 2 \left(2\pi \sqrt{\frac{m}{k}}\right) = 2T$. Thus, it doubles.

Explanation:Potential energy $PE = \frac{1}{2} k x^2$ and kinetic energy $KE = \frac{1}{2} k (A^2 - x^2)$.
Setting $PE = KE$:
$$\frac{1}{2} k x^2 = \frac{1}{2} k (A^2 - x^2) \implies 2x^2 = A^2 \implies x = \frac{A}{\sqrt{2}}$$

Explanation:Total mechanical energy is $E = \frac{1}{2} k A^2$. Since $E \propto A^2$, doubling the amplitude ($2A$) increases the total energy by $2^2 = 4$ times.

Explanation:By definition, a seconds pendulum is a pendulum whose period of oscillation is exactly 2 seconds (1 second for half-oscillation in each direction).

Explanation:Displacement is $x = A \sin(\omega t)$ and acceleration is $a = -\omega^2 A \sin(\omega t) = \omega^2 A \sin(\omega t + \pi)$. The phase difference is therefore $\pi\text{ rad}$ ($180^\circ$).

Explanation:Effective gravitational acceleration in an upward accelerating frame is $g_{eff} = g + a = g + \frac{g}{2} = \frac{3g}{2}$.
New period $T' = 2\pi \sqrt{\frac{L}{g_{eff}}} = 2\pi \sqrt{\frac{L}{\frac{3}{2}g}} = T \sqrt{\frac{2}{3}}$.

Explanation:Inside an orbiting satellite in free fall, the effective acceleration due to gravity is $g_{eff} = 0$.
$$T = 2\pi \sqrt{\frac{L}{0}} \to \infty$$
The pendulum will not oscillate at all.

Explanation:For springs connected in parallel, both undergo the same displacement. Total restoring force is $F = F_1 + F_2 = -k x - k x = -2kx$. Thus, equivalent force constant $k_{eq} = k + k = 2k$.

Explanation:For springs in series, equivalent constant is given by:
$$\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} = \frac{1}{100} + \frac{1}{100} = \frac{2}{100}$$
$$k_{eq} = \frac{100}{2} = 50\text{ N/m}$$

Explanation:In damped oscillations, dissipative forces (like viscous resistance or friction) continuously extract energy from the system, resulting in an exponential decay of amplitude $A(t) = A_0 e^{-bt/2m}$.

Explanation:Resonance is the condition in forced oscillations where the frequency of the external periodic driving force matches the natural frequency of the system, causing the amplitude of oscillation to reach a maximum value.

Explanation:Maximum velocity is $v_{max} = \omega A$ and maximum acceleration is $a_{max} = \omega^2 A$.
Dividing $a_{max}$ by $v_{max}$:
$$\omega = \frac{a_{max}}{v_{max}} = \frac{50}{10} = 5\text{ rad/s}$$

Explanation:Potential energy is $U = \frac{1}{2} m \omega^2 x^2$.
For $x = \frac{A}{2}$:
$$U = \frac{1}{2} m \omega^2 \left(\frac{A}{2}\right)^2 = \frac{1}{8} m \omega^2 A^2$$

Explanation:Time period is $T = 2\pi \sqrt{\frac{L}{g}}$.
With length $L' = 4L$:
$$T' = 2\pi \sqrt{\frac{4L}{g}} = 2 \left(2\pi \sqrt{\frac{L}{g}}\right) = 2T$$