301. An overdamped oscillator differs from a lightly damped oscillator because the overdamped oscillator
ⓐ. returns without repeated equilibrium crossings
ⓑ. oscillates with constant amplitude forever
ⓒ. loses the restoring tendency at all displacements
ⓓ. gains amplitude after each attempted cycle
Correct Answer: returns without repeated equilibrium crossings
Explanation: In an overdamped oscillator, damping is so strong that the system does not complete oscillations about the mean position. After being displaced and released, it returns slowly toward equilibrium without repeated crossings. A lightly damped oscillator still crosses the mean position again and again, but with decreasing amplitude. The restoring tendency is still present in an overdamped system, but the resistive effect is too large for oscillatory motion. Constant amplitude belongs to an ideal undamped oscillator, not to an overdamped one.
302. A displacement-time record shows that an oscillator crosses the mean position many times, but the peak displacement after each crossing is smaller than before. The motion is best classified as
ⓐ. undamped SHM
ⓑ. lightly damped oscillation
ⓒ. critically damped motion
ⓓ. overdamped non-oscillatory motion
Correct Answer: lightly damped oscillation
Explanation: Repeated crossings of the mean position show that the motion is still oscillatory. The decreasing peak displacement shows that the amplitude is reducing with time. This is the behavior of a lightly damped oscillator, where resistive forces remove mechanical energy gradually. Critical and overdamped cases return to equilibrium without repeated oscillations. Ideal undamped SHM would have equal peak displacements cycle after cycle.
303. A damped spring oscillator has its amplitude reduced from \(A_0\) to \(\frac{A_0}{2}\). If its mechanical energy is proportional to the square of amplitude, the remaining energy is
ⓐ. \(\frac{E_0}{2}\)
ⓑ. \(\frac{E_0}{4}\)
ⓒ. \(2E_0\)
ⓓ. \(4E_0\)
Correct Answer: \(\frac{E_0}{4}\)
Explanation: \( \textbf{Given:} \) Initial amplitude \(A_0\), later amplitude \(A=\frac{A_0}{2}\).
For a spring-type oscillator,
\[
E\propto A^2
\]
Therefore,
\[
\frac{E}{E_0}=\left(\frac{A}{A_0}\right)^2
\]
Substitute \(A=\frac{A_0}{2}\):
\[
\frac{E}{E_0}=\left(\frac{1}{2}\right)^2
\]
\[
\frac{E}{E_0}=\frac{1}{4}
\]
So,
\[
E=\frac{E_0}{4}
\]
Halving amplitude does not halve energy because energy depends on \(A^2\).
\( \textbf{Final answer:} \) \(E=\frac{E_0}{4}\).
304. Study the table comparing damped motion cases.
| Row | Damping case | Behavior |
|---|
| P | Light damping | Oscillates with decreasing amplitude |
| Q | Critical damping | Returns fastest without oscillation |
| R | Overdamping | Returns slowly without oscillation |
| S | Undamped ideal motion | Amplitude decreases due to resistive loss |
The row that is not suitable is
ⓐ. Row P
ⓑ. Row Q
ⓒ. Row R
ⓓ. Row S
Correct Answer: Row S
Explanation: In light damping, the oscillator still oscillates but the amplitude decreases. In critical damping, the return to equilibrium is the fastest possible without oscillation. In overdamping, the system also avoids oscillation, but it returns more slowly. Undamped ideal motion assumes no resistive loss, so the amplitude remains constant. Row S assigns damping behavior to an ideal undamped system.
305. A free oscillator is one that, after being disturbed and released, oscillates mainly with
ⓐ. its natural frequency
ⓑ. the frequency of an external driver only
ⓒ. zero frequency at all amplitudes
ⓓ. a frequency chosen independently of the system
Correct Answer: its natural frequency
Explanation: A free oscillator is set into motion and then left to itself, apart from its own restoring influence and inertia. In the ideal case, it oscillates with its natural frequency. The natural frequency is determined by system parameters such as \(k\) and \(m\) for a spring oscillator or \(l\) and \(g\) for a simple pendulum. A forced oscillator, not a free oscillator, is continuously driven by an external periodic agency. The word “natural” indicates the frequency selected by the system itself for those system parameters.
306. In forced oscillations, the oscillator is acted on by
ⓐ. a constant force only
ⓑ. a periodic external force
ⓒ. no external influence after release
ⓓ. a force always proportional to speed in the same direction
Correct Answer: a periodic external force
Explanation: Forced oscillation occurs when an external periodic force acts on a system capable of oscillating. The force supplies energy to the system at the driving frequency. After transients die out, the oscillator responds mainly at the frequency of the external driving force. This differs from free oscillation, where the system is disturbed and then left to oscillate at its natural frequency. A constant force may shift equilibrium, but it does not by itself produce forced oscillations.
307. A child on a swing is pushed periodically. The amplitude grows large when the pushes are timed so that their frequency is close to the swing’s natural frequency. This is an example of
ⓐ. critical damping
ⓑ. resonance
ⓒ. uniform circular motion
ⓓ. overdamping
Correct Answer: resonance
Explanation: Resonance occurs when a system is driven by an external periodic force whose frequency is close to the system’s natural frequency. Under this condition, energy transfer from the driver to the oscillator becomes especially effective. A swing pushed at the right rhythm can therefore reach a large amplitude. Critical damping and overdamping describe non-oscillatory return to equilibrium, not large-amplitude forced motion. The timing of the driving force is the key feature in resonance.
308. In steady forced oscillation, the frequency of the oscillator is mainly equal to
ⓐ. the driving frequency
ⓑ. zero at all times
ⓒ. twice the amplitude
ⓓ. the damping constant only
Correct Answer: the driving frequency
Explanation: In forced oscillation, an external periodic force continuously drives the system. After the initial transient part dies away, the oscillator settles into steady motion at the frequency of the driving force. The natural frequency still matters because it affects the amplitude of response. The largest response occurs when the driving frequency is close to the natural frequency. Frequency is a timing property, so it is not determined by amplitude alone.
309. A driven oscillator has natural frequency \(f_0=5\,\text{Hz}\). It is driven successively by external forces of frequencies \(2\,\text{Hz}\), \(4.8\,\text{Hz}\), \(8\,\text{Hz}\), and \(12\,\text{Hz}\). The largest amplitude is expected near
ⓐ. \(2.0\,\text{Hz}\)
ⓑ. \(4.8\,\text{Hz}\)
ⓒ. \(8.0\,\text{Hz}\)
ⓓ. \(12.0\,\text{Hz}\)
Correct Answer: \(4.8\,\text{Hz}\)
Explanation: Resonance occurs when the driving frequency is close to the natural frequency of the oscillator.
\( \textbf{Natural frequency:} \)
\[
f_0=5\,\text{Hz}
\]
Compare the given driving frequencies with \(5\,\text{Hz}\).
The differences are:
\[
|2-5|=3\,\text{Hz}
\]
\[
|4.8-5|=0.2\,\text{Hz}
\]
\[
|8-5|=3\,\text{Hz}
\]
\[
|12-5|=7\,\text{Hz}
\]
The closest driving frequency is \(4.8\,\text{Hz}\).
The amplitude is expected to be largest near this value, especially if damping is small.
\( \textbf{Final answer:} \) \(4.8\,\text{Hz}\).
310. Use the graph description below.
The amplitude of a driven oscillator is plotted against driving frequency. The curve has a peak near the natural frequency of the oscillator.
The peak of the curve represents
ⓐ. maximum response due to resonance
ⓑ. zero energy transfer from the driver
ⓒ. motion with no restoring force
ⓓ. overdamped return without oscillation
Correct Answer: maximum response due to resonance
Explanation: A plot of amplitude against driving frequency for a forced oscillator is called a resonance response curve. The peak indicates that the oscillator responds with maximum amplitude for a driving frequency close to its natural frequency. At this condition, energy supplied by the driver is transferred efficiently to the oscillator. Damping controls how high and sharp the peak is. The peak does not represent absence of restoring force; it represents strong forced response.
311. If damping in a driven oscillator is increased, the resonance peak generally becomes
ⓐ. higher and sharper
ⓑ. lower and broader
ⓒ. infinitely high
ⓓ. exactly unchanged
Correct Answer: lower and broader
Explanation: Damping removes mechanical energy from an oscillator. In a driven oscillator, stronger damping prevents the amplitude from growing very large at resonance. The resonance peak therefore becomes lower. It also becomes broader, meaning the response is spread over a wider range of driving frequencies. Weak damping gives a sharper and higher resonance peak, while strong damping makes resonance less pronounced.
312. A lightly damped oscillator is driven at resonance. The amplitude does not become infinite in a real system because
ⓐ. damping dissipates energy as it is supplied
ⓑ. the restoring force vanishes at all displacements
ⓒ. the natural frequency becomes zero
ⓓ. the driving force stops being periodic
Correct Answer: damping dissipates energy as it is supplied
Explanation: In an ideal undamped model, resonance can lead to very large amplitude because energy keeps accumulating. Real oscillators have damping due to friction, air resistance, or internal losses. At resonance, the driver supplies energy efficiently, but damping also removes energy continuously. A steady amplitude is reached when the energy supplied per cycle balances the energy dissipated per cycle. This is why real resonance peaks are finite rather than infinite.
313. The following statements refer to forced oscillations and resonance.
I. A forced oscillator is driven by an external periodic force.
II. Resonance occurs when the driving frequency is close to the natural frequency.
III. Greater damping usually makes the resonance peak sharper and higher.
Select the valid set.
ⓐ. I only
ⓑ. I and II only
ⓒ. II and III only
ⓓ. I, II, and III
Correct Answer: I and II only
Explanation: Statement I is valid because forced oscillation requires an external periodic driving force. Statement II is also valid because resonance is the large response obtained when the driving frequency is close to the natural frequency. Statement III is not valid because greater damping usually reduces the height of the resonance peak and makes it broader. A sharp high peak is associated with weak damping. Damping controls how strongly the oscillator can build up amplitude under repeated driving.
314. A machine part vibrates strongly when a motor runs at a particular speed. The most likely reason is that the motor’s driving frequency is
ⓐ. close to the natural frequency of the part
ⓑ. exactly zero
ⓒ. unrelated to the vibration frequency
ⓓ. smaller than every possible natural frequency by definition
Correct Answer: close to the natural frequency of the part
Explanation: Strong vibration of a part at a particular motor speed is a typical sign of resonance. The rotating motor can provide a periodic driving force. If the driving frequency becomes close to the natural frequency of the machine part, the part can absorb energy efficiently and vibrate with large amplitude. Changing the motor speed away from that frequency usually reduces the vibration. This is why resonance must be considered in the design of machines and structures.
315. A bridge is driven by repeated periodic forces. To reduce the risk of resonance, engineers try to ensure that the driving frequencies are
ⓐ. close to the bridge’s natural frequencies
ⓑ. always larger than the speed of sound
ⓒ. equal to zero while the bridge is in use
ⓓ. far from the bridge’s natural frequencies
Correct Answer: far from the bridge’s natural frequencies
Explanation: Resonance can produce large amplitudes when a structure is driven near one of its natural frequencies. For a bridge, repeated forces due to wind, traffic, or marching can act periodically. If these frequencies match or come close to a natural frequency, the vibration amplitude can become large. To avoid this, design and operating conditions aim to keep common driving frequencies away from natural frequencies. The goal is not to remove all forces, but to avoid efficient resonant energy transfer.
316. Match the term with its most suitable meaning.
| Column I | Column II |
|---|
| P. Free oscillation | 1. Motion mainly at the system’s natural frequency after release |
| Q. Forced oscillation | 2. Oscillation maintained by an external periodic force |
| R. Resonance | 3. Large response when driving frequency is close to natural frequency |
| S. Damping | 4. Gradual loss of mechanical energy due to resistive effects |
ⓐ. P-4, Q-3, R-2, S-1
ⓑ. P-2, Q-1, R-4, S-3
ⓒ. P-3, Q-2, R-1, S-4
ⓓ. P-1, Q-2, R-3, S-4
Correct Answer: P-1, Q-2, R-3, S-4
Explanation: Free oscillation occurs when a system is disturbed and then left to oscillate mainly at its natural frequency. Forced oscillation is maintained by an external periodic driving force. Resonance is the large response obtained when the driving frequency is close to the natural frequency. Damping is the loss of mechanical energy due to resistive effects, causing amplitude to decrease. These terms describe different parts of the same broad study of oscillatory motion.
317. A driven oscillator has weak damping in Case 1 and strong damping in Case 2. Both are driven near their natural frequencies with the same driving force. The expected comparison of maximum amplitudes is
ⓐ. damping has no effect on maximum amplitude
ⓑ. Case 2 has a larger maximum amplitude
ⓒ. both must have zero maximum amplitude
ⓓ. Case 1 has a larger maximum amplitude
Correct Answer: Case 1 has a larger maximum amplitude
Explanation: Near resonance, energy transfer from the driving force to the oscillator is efficient. Weak damping removes less energy per cycle, so the oscillator can build up a larger amplitude. Strong damping removes energy more rapidly, limiting the amplitude. Therefore, the weakly damped oscillator has the larger resonance response. The driving force may be the same, but the balance between supplied energy and dissipated energy is different.
318. A forced oscillator is driven at angular frequency \(\omega_d\), while its natural angular frequency is \(\omega_0\). The resonance condition is approximately
ⓐ. \(\omega_d\approx\omega_0\)
ⓑ. \(\omega_d=0\)
ⓒ. \(\omega_d\approx\frac{1}{\omega_0}\)
ⓓ. \(\omega_d\gg\omega_0\) for every oscillator
Correct Answer: \(\omega_d\approx\omega_0\)
Explanation: Resonance occurs when the driving frequency is close to the natural frequency of the oscillator. In angular-frequency notation, this is written approximately as \(\omega_d\approx\omega_0\). Under this condition, the driver supplies energy at the rhythm most effective for building up the oscillation. Damping may shift and broaden the practical peak, but the basic Class 11 resonance idea is closeness between driving and natural frequencies. The condition is not based on reciprocal frequency or an arbitrarily very large driving frequency.
319. Use the arrangement described below. A point moves uniformly around a circle of radius \(A\) with angular speed \(\omega\). The shadow of the point on a horizontal diameter is observed. The motion of the shadow is
ⓐ. uniform circular motion with radius \(A\)
ⓑ. motion that never repeats
ⓒ. motion with constant speed along the diameter
ⓓ. simple harmonic motion along the diameter
Correct Answer: simple harmonic motion along the diameter
Explanation: The projection of uniform circular motion on a diameter performs simple harmonic motion. If the rotating point has angular speed \(\omega\), the projected displacement can be written as \(x=A\cos\omega t\) or \(x=A\sin\omega t\), depending on the starting point. The projection moves to and fro between \(+A\) and \(-A\). Although the point on the circle has constant speed, the projected shadow does not have constant speed along the diameter. This projection model helps connect SHM phase with uniform angular motion.
320. A particle moves uniformly on a circle of radius \(A\), and its projection on a diameter has displacement \(x=A\cos\omega t\). The amplitude of the projected SHM is
ⓐ. \(2\pi A\)
ⓑ. \(\omega A\)
ⓒ. \(\omega^2A\)
ⓓ. \(A\)
Correct Answer: \(A\)
Explanation: In the projection model, the circular radius is equal to the maximum possible displacement of the projection from the centre. The displacement equation \(x=A\cos\omega t\) has coefficient \(A\) outside the cosine function. Since the cosine value ranges from \(-1\) to \(+1\), the projected displacement ranges from \(-A\) to \(+A\). Therefore, the amplitude of the SHM is \(A\). The quantities \(\omega A\) and \(\omega^2A\) correspond to maximum speed and maximum acceleration magnitude, not amplitude.
