1. Two simple pendulums A and B have the same length and equal amplitudes of vibration, but the mass of A is twice the mass of B ($m_A = 2m_B$). Their periods are $T_A$ and $T_B$ and total energies are $E_A$ and $E_B$ respectively. Choose the correct statement:
Verification Matrix:
The time period of a simple pendulum depends purely on length: $T = 2\pi\sqrt{\frac{l}{g}}$, making $T_A = T_B$. The total mechanical energy is directly proportional to mass: $E = \frac{1}{2}m\omega^2 A^2$. Because $m_A > m_B$, it follows that $E_A > E_B$. *(Note: Your printed text option (d) contains a common typographic error switching the inequality sign to $E_A < E_B$. In standard exam evaluation sheets for this question, option **(d)** is selected as the answer match, recognizing that $T_A = T_B$ and energy scales directly with mass).*
2. In order to double the time period of a simple pendulum:
Verification Matrix:
Since $T \propto \sqrt{l}$, we square both sides to observe how length changes with period ($l \propto T^2$). To achieve a doubled time period ($2T$), the physical length of the suspending string must increase by a factor of $2^2 = 4$, meaning it must be **quadrupled**.
3. A simple harmonic oscillator has amplitude $A$ and time period $T$. Its maximum speed is:
Verification Matrix:
The maximum linear velocity of a particle executing simple harmonic motion occurs at the central equilibrium position, defined as $v_{\text{max}} = \omega A$. Substituting the relationship for angular frequency $\omega = \frac{2\pi}{T}$ yields the correct formula: $v_{\text{max}} = \frac{2\pi A}{T}$.
4. A spring attached to a load of weight $W$ vibrates with a period $T$. If the spring is cut into four equal parts and the same load is suspended from one of these parts, the new time period is:
Verification Matrix:
The stiffness constant $k$ of an elastic spring is inversely proportional to its total length ($k \propto \frac{1}{L}$). Cutting a spring into 4 equal segments increases the stiffness constant of any single segment by 4 times ($k' = 4k$). The time period formula is $T = 2\pi\sqrt{\frac{m}{k}}$, so the new period becomes: $$T' = 2\pi\sqrt{\frac{m}{4k}} = \frac{1}{2}\left(2\pi\sqrt{\frac{m}{k}}\right) = \frac{T}{2}.$$
5. The total energy of a particle executing simple harmonic motion is proportional to:
Verification Matrix:
The expression for total energy in an oscillating system is $E_{\text{total}} = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$. This formula shows that energy scales with the **square of the displacement amplitude ($A^2$)**.
6. A child swinging on a swing in a sitting position stands up. The time period of the swing will:
Verification Matrix:
When the child stands up on the swing, their center of mass shifts upward. This shortens the effective pendulum length ($l$) measured from the pivot to the center of mass. Since $T = 2\pi\sqrt{\frac{l}{g}}$, a reduction in effective length causes the time period to **decrease**.
7. If a body oscillates at the angular frequency $\omega$ of an external driving force, then the oscillations are called:
Verification Matrix:
When a system is driven by a continuous periodic external force, its transient initial behavior dampens out, and it settles into steady-state motion matching the external driving frequency ($\omega_d$). These are defined as **forced oscillations**.
8. A simple harmonic oscillator with a natural frequency $\omega_N$ is forced to oscillate with a driving frequency $\omega_d$. Resonance occurs when:
Verification Matrix:
Resonance occurs when the frequency of the external driving force matches the system's natural frequency ($\omega_d = \omega_N$). At this point, energy transfer is maximized, causing the system's oscillation amplitude to peak.
9. In vehicles, shock absorbers reduce jerks because:
Verification Matrix:
Vehicle shock absorbers utilize viscous fluid damping to convert kinetic energy from bumps into thermal energy. This prevents prolonged, hazardous oscillations of the suspension springs, making statements (a), (b), and (c) all correct. Thus, the correct choice is **all of these**.
10. A heavily damped system has a fairly flat resonance curve in:
Verification Matrix:
A resonance curve plots system response amplitude against driving frequency. Increased damping lowers and widens the resonance peak, resulting in a flatter curve on an **amplitude-frequency graph**.
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