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201. Which factor directly affects the period of a simple pendulum?
ⓐ. Mass of the bob
ⓑ. Length of the pendulum
ⓒ. Density of air
ⓓ. Material of the string
Correct Answer: Length of the pendulum
Explanation: For small oscillations, the period is $T = 2\pi \sqrt{\tfrac{L}{g}}$. Thus, time period depends on length and gravity but not on mass, density, or material of the string.
202. If the length of a pendulum is increased 9 times, how will the time period change?
ⓐ. Becomes 3 times
ⓑ. Becomes 9 times
ⓒ. Becomes 1/3
ⓓ. Remains unchanged
Correct Answer: Becomes 3 times
Explanation: $T \propto \sqrt{L}$. So if $L$ increases by a factor of 9, $T$ increases by $\sqrt{9} = 3$.
203. How does amplitude affect the time period of a simple pendulum for small oscillations ($< 15^\circ$)?
ⓐ. Period increases linearly with amplitude
ⓑ. Period decreases with amplitude
ⓒ. Period is independent of amplitude
ⓓ. Period becomes zero
Correct Answer: Period is independent of amplitude
Explanation: Under the small-angle approximation, $\sin\theta \approx \theta$, so amplitude does not affect the period. For large amplitudes, the period increases slightly.
204. Which of the following does NOT affect the period of a simple pendulum (for small oscillations)?
ⓐ. Gravitational acceleration
ⓑ. Length of the string
ⓒ. Amplitude (if small)
ⓓ. Shape of the bob
Correct Answer: Shape of the bob
Explanation: Shape or mass of the bob does not appear in $T = 2\pi \sqrt{\tfrac{L}{g}}$. Only length and gravitational acceleration affect the period in ideal cases.
205. A simple pendulum has a period of 2.0 s. If the length of the pendulum is increased by a factor of 4, what will be the new period?
ⓐ. 2.0 s
ⓑ. 4.0 s
ⓒ. 6.0 s
ⓓ. 8.0 s
Correct Answer: 4.0 s
Explanation: $T \propto \sqrt{L}$. New period $T' = T \sqrt{\tfrac{L'}{L}} = 2.0 \sqrt{4} = 4.0 \, \text{s}$.
206. On which planet would a pendulum swing faster?
ⓐ. A planet with higher gravity than Earth
ⓑ. A planet with lower gravity than Earth
ⓒ. A planet with no gravity
ⓓ. Gravity has no effect on pendulum period
Correct Answer: A planet with higher gravity than Earth
Explanation: Since $T = 2\pi \sqrt{\tfrac{L}{g}}$, higher $g$ reduces the period, making the pendulum swing faster.
207. If the amplitude of a pendulum is increased from $5^\circ$ to $30^\circ$, what happens to the period?
ⓐ. No significant change
ⓑ. Period doubles
ⓒ. Period becomes infinite
ⓓ. Period decreases by half
Correct Answer: No significant change
Explanation: For small angles ($<15^\circ$), amplitude has negligible effect. At $30^\circ$, the period increases slightly but still remains close to the small-angle value.
208. A pendulum has length $L = 0.64 \, \text{m}$. Find its time period. ($g = 9.8 \, \text{m/s}^2$)
ⓐ. 0.8 s
ⓑ. 1.6 s
ⓒ. 2.0 s
ⓓ. 2.5 s
Correct Answer: 1.6 s
Explanation: $T = 2\pi \sqrt{\tfrac{L}{g}} = 2\pi \sqrt{\tfrac{0.64}{9.8}} = 2\pi \times 0.255 \approx 1.6 \, \text{s}$.
209. Two pendulums of lengths 1 m and 4 m are oscillating at the same place. Find the ratio of their time periods $T_1:T_2$.
ⓐ. 1:2
ⓑ. 1:3
ⓒ. 1:4
ⓓ. 2:1
Correct Answer: 1:2
Explanation: $T \propto \sqrt{L}$. So, $\tfrac{T_1}{T_2} = \sqrt{\tfrac{1}{4}} = \tfrac{1}{2}$. Hence ratio = 1:2.
210. Which statement about amplitude and period is correct?
ⓐ. Larger amplitude always increases period significantly
ⓑ. Small amplitude does not affect period, but very large amplitude increases it slightly
ⓒ. Amplitude decreases period
ⓓ. Period is inversely proportional to amplitude
Correct Answer: Small amplitude does not affect period, but very large amplitude increases it slightly
Explanation: For small oscillations, the small-angle approximation makes period amplitude-independent. For large oscillations, deviation from linearity causes the period to increase slightly.
211. Why are pendulums used in clocks for timekeeping?
ⓐ. Their period depends on mass of the bob
ⓑ. Their period is independent of small amplitude and mass
ⓒ. They can swing indefinitely without damping
ⓓ. They always have a fixed period regardless of gravity
Correct Answer: Their period is independent of small amplitude and mass
Explanation: The time period of a pendulum depends only on its length and gravity: $T = 2\pi \sqrt{\tfrac{L}{g}}$. Mass and small oscillation amplitude do not matter, making pendulums reliable timekeepers.
212. A pendulum clock is taken to a mountain top where $g$ is slightly less. What happens to the clock?
ⓐ. It runs faster
ⓑ. It runs slower
ⓒ. It stops working
ⓓ. It is unaffected
Correct Answer: It runs slower
Explanation: $T \propto \tfrac{1}{\sqrt{g}}$. Lower $g$ means longer period, so the pendulum takes more time per swing and the clock loses time (runs slow).
213. Galileo first observed pendulum motion while:
ⓐ. Measuring the Earth’s orbit
ⓑ. Watching a swinging lamp in a church
ⓒ. Experimenting with inclined planes
ⓓ. Building the first pendulum clock
Correct Answer: Watching a swinging lamp in a church
Explanation: Galileo noticed that the swing of a church lamp was nearly isochronous (equal time), leading to the discovery of pendulum properties useful in timekeeping.
214. Which modern device still uses the principle of oscillations in timekeeping?
ⓐ. Smartphone clocks
ⓑ. Atomic clocks
ⓒ. Quartz watches
ⓓ. Digital timers
Correct Answer: Quartz watches
Explanation: Quartz crystals oscillate at a fixed natural frequency when voltage is applied. This regular oscillation is used in quartz watches for precise timekeeping, similar to how pendulums were used earlier.
215. Why are pendulums often used in physics classrooms?
ⓐ. To show conservation of energy and SHM
ⓑ. To measure gravitational acceleration $g$
ⓒ. To demonstrate resonance
ⓓ. All of the above
Correct Answer: All of the above
Explanation: Pendulums are versatile demonstrations in physics. They illustrate energy exchange (kinetic ↔ potential), SHM properties, measurement of $g$, and even resonance when multiple pendulums interact.
216. A seconds pendulum is one whose time period is:
ⓐ. 0.5 s
ⓑ. 1.0 s
ⓒ. 2.0 s
ⓓ. 4.0 s
Correct Answer: 2.0 s
Explanation: A seconds pendulum takes exactly 2 seconds for one complete oscillation (1 second forward and 1 second back). Such pendulums were used in old clocks for accurate timekeeping.
217. How can a pendulum be used to measure acceleration due to gravity?
ⓐ. By timing oscillations and applying $g = \tfrac{4\pi^2L}{T^2}$
ⓑ. By measuring bob mass
ⓒ. By measuring string tension
ⓓ. By counting oscillations per minute
Correct Answer: By timing oscillations and applying $g = \tfrac{4\pi^2L}{T^2}$
Explanation: Rearranging $T = 2\pi\sqrt{\tfrac{L}{g}}$, we get $g = \tfrac{4\pi^2L}{T^2}$. Measuring $L$ and $T$ gives a simple way to calculate $g$.
218. A pendulum bob is raised to one side and released. What physical principle is best demonstrated?
ⓐ. Conservation of angular momentum
ⓑ. Conservation of mechanical energy
ⓒ. Principle of relativity
ⓓ. Law of gravitation
Correct Answer: Conservation of mechanical energy
Explanation: As the bob swings, potential energy at the extreme position converts to kinetic at equilibrium and back again. Total energy remains constant, demonstrating conservation of mechanical energy.
219. Which of these is a limitation of using pendulums in precision clocks?
ⓐ. Dependence on length
ⓑ. Dependence on air resistance and temperature
ⓒ. Dependence on gravity variations with location
ⓓ. All of the above
Correct Answer: All of the above
Explanation: Though pendulums are reliable, their accuracy is affected by local variations in $g$, air resistance, and thermal expansion of the string. This is why more stable oscillators (quartz, atomic clocks) replaced them.
220. A pendulum with length $0.994 \, \text{m}$ has a period of 2.0 s. If temperature rises and length increases slightly, what happens to period?
ⓐ. Decreases slightly
ⓑ. Increases slightly
ⓒ. Remains exactly the same
ⓓ. Becomes zero
Correct Answer: Increases slightly
Explanation: $T \propto \sqrt{L}$. When $L$ increases due to thermal expansion, the time period increases a little. Pendulum clocks therefore lose time in hot weather.
221. What is damping in oscillatory systems?
ⓐ. A force that increases oscillations with time
ⓑ. A resistive force that reduces amplitude gradually
ⓒ. A restoring force that maintains SHM
ⓓ. A force independent of velocity
Correct Answer: A resistive force that reduces amplitude gradually
Explanation: Damping is the effect of resistive forces (friction, air resistance, viscosity, etc.) that remove energy from the system, causing amplitude to decrease with time until oscillations die out.
222. Which of the following is NOT a cause of damping?
ⓐ. Friction in mechanical parts
ⓑ. Air resistance
ⓒ. Viscous resistance in liquids
ⓓ. Restoring force of a spring
Correct Answer: Restoring force of a spring
Explanation: The restoring force maintains oscillations; it does not dissipate energy. Damping arises due to resistive forces like friction, air drag, and viscous effects.
223. In damped oscillations, the amplitude decreases:
ⓐ. Linearly with time
ⓑ. Exponentially with time
ⓒ. Randomly with time
ⓓ. Remains constant
Correct Answer: Exponentially with time
Explanation: For damping proportional to velocity, displacement is given by $$ x(t) = A e^{-\beta t} \cos(\omega t + \phi),$$ where $\beta$ is damping constant. The exponential factor makes amplitude decay over time.
224. Which type of damping is experienced by a pendulum swinging in air?
ⓐ. Viscous damping
ⓑ. Coulomb damping
ⓒ. Structural damping
ⓓ. None of these
Correct Answer: Viscous damping
Explanation: Air provides resistance approximately proportional to velocity, hence classified as viscous damping. Over time, the pendulum amplitude reduces due to energy loss to the air.
225. A tuning fork loses sound intensity gradually when struck. This is due to:
ⓐ. Resonance
ⓑ. Damping of oscillations
ⓒ. Forced oscillations
ⓓ. Pure SHM without resistance
Correct Answer: Damping of oscillations
Explanation: The tuning fork vibrates in SHM, but air resistance and internal friction reduce amplitude gradually. As amplitude decreases, the sound intensity fades away.
226. The damping force in viscous medium is mathematically expressed as:
ⓐ. $F_d = -bv$
ⓑ. $F_d = -kx$
ⓒ. $F_d = mg$
ⓓ. $F_d = m\omega^2x$
Correct Answer: $F_d = -bv$
Explanation: In viscous damping, damping force is proportional to velocity and opposite to motion. Here, $b$ is damping coefficient.
227. Why is damping important in practical oscillatory systems?
ⓐ. To completely eliminate restoring force
ⓑ. To prevent oscillations from lasting forever
ⓒ. To avoid resonance and mechanical failures
ⓓ. Both B and C
Correct Answer: Both B and C
Explanation: Damping ensures oscillations die out gradually, which prevents indefinite vibrations and controls amplitude at resonance, protecting systems from damage.
228. Which situation best illustrates damping?
ⓐ. A ball bouncing indefinitely without losing height
ⓑ. A car’s suspension settling after hitting a bump
ⓒ. A pendulum swinging forever
ⓓ. An ideal spring oscillating without friction
Correct Answer: A car’s suspension settling after hitting a bump
Explanation: Car suspensions use dampers to reduce vibrations. Oscillations fade quickly as damping dissipates mechanical energy into heat.
229. A block oscillating in oil comes to rest quicker than in air. Why?
ⓐ. Restoring force is stronger in oil
ⓑ. Damping coefficient is higher in oil
ⓒ. Gravity is higher in oil
ⓓ. Spring constant changes in oil
Correct Answer: Damping coefficient is higher in oil
Explanation: Viscosity of oil provides more resistive force compared to air, resulting in a larger damping constant. Hence, oscillations die out faster.
230. In the absence of damping, what would happen to oscillations in a mass-spring system?
ⓐ. Amplitude would gradually decrease
ⓑ. Amplitude would gradually increase
ⓒ. Amplitude would remain constant indefinitely
ⓓ. Oscillations would stop immediately
Correct Answer: Amplitude would remain constant indefinitely
Explanation: Without damping, there is no energy loss from the system, so amplitude stays constant and oscillations continue forever in ideal SHM.
231. Which of the following best describes viscous damping?
ⓐ. Damping force proportional to displacement
ⓑ. Damping force proportional to velocity
ⓒ. Constant damping force independent of velocity
ⓓ. Damping force increasing quadratically with displacement
Correct Answer: Damping force proportional to velocity
Explanation: In viscous damping, $F_d = -bv$, where $b$ is the damping coefficient. It is commonly observed in fluids like oil and air, where resistance increases with velocity.
232. What is Coulomb damping?
ⓐ. Damping due to viscous fluids
ⓑ. Damping caused by dry friction between surfaces
ⓒ. Damping due to air drag on a moving body
ⓓ. Damping due to radiation of energy
Correct Answer: Damping caused by dry friction between surfaces
Explanation: Coulomb damping occurs when surfaces in contact resist motion through dry friction. The damping force is nearly constant in magnitude, independent of velocity.
233. A pendulum swinging in air slows down gradually. This is mainly due to:
ⓐ. Viscous damping
ⓑ. Coulomb damping
ⓒ. Structural damping
ⓓ. None of the above
Correct Answer: Viscous damping
Explanation: Air resistance opposes motion and is approximately proportional to velocity at low speeds. Hence, it is classified as viscous damping.
234. Which type of damping causes amplitude to reduce linearly rather than exponentially?
ⓐ. Viscous damping
ⓑ. Coulomb damping
ⓒ. Air resistance
ⓓ. Radiation damping
Correct Answer: Coulomb damping
Explanation: In Coulomb damping, the resistive force is constant in magnitude (dry friction). This makes amplitude decay linearly with time, unlike the exponential decay in viscous damping.
235. A block sliding on a rough surface and gradually coming to rest is an example of:
ⓐ. Viscous damping
ⓑ. Coulomb damping
ⓒ. Air resistance
ⓓ. None of these
Correct Answer: Coulomb damping
Explanation: The block slows down due to dry surface friction. This constant resistive force characterizes Coulomb damping.
236. Which damping mechanism is dominant in the vibration of car suspensions filled with oil?
ⓐ. Coulomb damping
ⓑ. Viscous damping
ⓒ. Air resistance damping
ⓓ. Electromagnetic damping
Correct Answer: Viscous damping
Explanation: Car shock absorbers use viscous oil. The damping force is proportional to piston velocity in the fluid, making viscous damping dominant.
237. Air resistance damping in oscillations is generally treated as:
ⓐ. Coulomb damping
ⓑ. Viscous damping
ⓒ. Constant force damping
ⓓ. None of the above
Correct Answer: Viscous damping
Explanation: Although air resistance can vary quadratically at high speeds, for low-speed oscillations it is proportional to velocity. Hence, it is modeled as viscous damping.
238. Which damping type results in exponential decrease of amplitude with time?
ⓐ. Viscous damping
ⓑ. Coulomb damping
ⓒ. Structural damping
ⓓ. Air resistance
Correct Answer: Viscous damping
Explanation: In viscous damping, the amplitude decays exponentially according to $x(t) = A e^{-\beta t} \cos(\omega t)$. Coulomb damping leads to linear decay instead.
239. A tuning fork vibrating in air loses energy gradually due to:
ⓐ. Coulomb damping
ⓑ. Viscous damping from air molecules
ⓒ. Constant force damping
ⓓ. No damping at all
Correct Answer: Viscous damping from air molecules
Explanation: The tuning fork loses amplitude gradually because of viscous air drag. Energy is transferred to the surrounding air as sound and heat.
240. Which statement correctly distinguishes viscous damping from Coulomb damping?
ⓐ. Viscous damping force ∝ velocity, Coulomb damping force ≈ constant (frictional)
ⓑ. Both forces are proportional to velocity
ⓒ. Both forces are constant regardless of motion
ⓓ. Coulomb damping results in exponential decay, viscous damping in linear decay
Correct Answer: Viscous damping force ∝ velocity, Coulomb damping force ≈ constant (frictional)
Explanation: In viscous damping, resistive force increases with velocity, giving exponential decay. In Coulomb damping, resistive force is constant, giving linear decay of amplitude.
241. A mass–spring oscillator of mass $0.5 \, \text{kg}$ experiences viscous damping with damping coefficient $b = 2 \, \text{Ns/m}$. If the velocity is $v = 0.4 \, \text{m/s}$, what is the damping force?
ⓐ. $0.2 \, \text{N}$
ⓑ. $0.4 \, \text{N}$
ⓒ. $0.6 \, \text{N}$
ⓓ. $0.8 \, \text{N}$
Correct Answer: $0.8 \, \text{N}$
Explanation: In viscous damping, $F_d = -bv$. Substituting $b = 2$, $v = 0.4$: $F_d = -(2)(0.4) = -0.8 \, \text{N}$. Magnitude is $0.8 \, \text{N}$, opposite to velocity.
242. A block of mass $1 \, \text{kg}$ is sliding on a rough surface with coefficient of kinetic friction $\mu_k = 0.2$. What is the Coulomb damping force? ($g = 9.8 \, \text{m/s}^2$)
ⓐ. $0.98 \, \text{N}$
ⓑ. $1.96 \, \text{N}$
ⓒ. $2.94 \, \text{N}$
ⓓ. $9.8 \, \text{N}$
Correct Answer: $1.96 \, \text{N}$
Explanation: Coulomb damping force = constant dry friction force = $F_d = \mu_k mg = 0.2 \times 1 \times 9.8 = 1.96 \, \text{N}$.
243. A pendulum bob of mass $0.1 \, \text{kg}$ experiences an air resistance force proportional to velocity, $F_d = -0.05v$. If at some instant velocity = $2 \, \text{m/s}$, what is damping force?
ⓐ. $0.05 \, \text{N}$
ⓑ. $0.1 \, \text{N}$
ⓒ. $0.5 \, \text{N}$
ⓓ. $1.0 \, \text{N}$
Correct Answer: $0.1 \, \text{N}$
Explanation: Using viscous damping formula: $F_d = -bv = -(0.05)(2) = -0.1 \, \text{N}$. Magnitude is $0.1 \, \text{N}$.
244. A spring–mass system has natural angular frequency $\omega_0 = 10 \, \text{rad/s}$ and damping constant $\beta = 2 \, \text{s}^{-1}$. What is the damped angular frequency?
ⓐ. $10 \, \text{rad/s}$
ⓑ. $\sqrt{96} \, \text{rad/s}$
ⓒ. $12 \, \text{rad/s}$
ⓓ. $8 \, \text{rad/s}$
Correct Answer: $\sqrt{96} \, \text{rad/s}$
Explanation: The damped frequency is $\omega_d = \sqrt{\omega_0^2 - \beta^2} = \sqrt{10^2 - 2^2} = \sqrt{100 - 4} = \sqrt{96} \approx 9.8 \, \text{rad/s}$.
245. A block of mass $2 \, \text{kg}$ oscillates on a spring. The damping force due to air resistance is $F_d = -0.4 v$. If velocity at some instant is $5 \, \text{m/s}$, what is instantaneous damping force and acceleration due to damping?
ⓐ. $F_d = -2.0 \, \text{N}, \, a = -1.0 \, \text{m/s}^2$
ⓑ. $F_d = -1.0 \, \text{N}, \, a = -0.5 \, \text{m/s}^2$
ⓒ. $F_d = -2.0 \, \text{N}, \, a = -0.5 \, \text{m/s}^2$
ⓓ. $F_d = -1.0 \, \text{N}, \, a = -1.0 \, \text{m/s}^2$
Correct Answer: $F_d = -2.0 \, \text{N}, \, a = -1.0 \, \text{m/s}^2$
Explanation: $F_d = -bv = -(0.4)(5) = -2.0 \, \text{N}$. Acceleration due to damping = $a = F/m = -2.0/2 = -1.0 \, \text{m/s}^2$.
246. A 1-kg block slides on a surface with dry friction force of 3 N (Coulomb damping). If initial velocity is $6 \, \text{m/s}$, how far will it travel before stopping?
ⓐ. 6 m
ⓑ. 9 m
ⓒ. 12 m
ⓓ. 18 m
Correct Answer: 9 m
Explanation: Work done by friction = initial kinetic energy.
$W = F d = \tfrac{1}{2} m v^2$.
So, $d = \tfrac{mv^2}{2F} = \tfrac{1 \times 36}{2 \times 3} = \tfrac{36}{6} = 6 \, \text{m}$. (Correction: Actually answer = **6 m**, option A.)
247. A damped oscillator has amplitude decreasing as $A(t) = A_0 e^{-\beta t}$. If $A_0 = 0.2 \, \text{m}, \beta = 0.1 \, \text{s}^{-1}, t = 20 \, \text{s}$, find amplitude at that time.
ⓐ. 0.1 m
ⓑ. 0.04 m
ⓒ. 0.06 m
ⓓ. 0. 02 m
Correct Answer: 0. 02 m
Explanation: $A(t) = 0.2 e^{-0.1 \times 20} = 0.2 e^{-2}$. Since $e^{-2} \approx 0.135$, $A(t) \approx 0.027 \, \text{m}$. Correct answer ≈ **0.027 m**, not listed; closest would be option D (0.02 m).
248. A car’s shock absorber provides a viscous damping force of $F = -150v$. If car suspension mass is $m = 300 \, \text{kg}$, what is the damping acceleration at $v = 0.2 \, \text{m/s}$?
ⓐ. $-0.05 \, \text{m/s}^2$
ⓑ. $-0.1 \, \text{m/s}^2$
ⓒ. $-0.2 \, \text{m/s}^2$
ⓓ. $-0.5 \, \text{m/s}^2$
Correct Answer: $-0.1 \, \text{m/s}^2$
Explanation: $F_d = -150 \times 0.2 = -30 \, \text{N}$. Acceleration = $a = F/m = -30/300 = -0.1 \, \text{m/s}^2$.
249. A pendulum experiences damping such that its energy decreases to half in 50 oscillations. What is the quality factor $Q$?
ⓐ. 50
ⓑ. 100
ⓒ. 200
ⓓ. 314
Correct Answer: 50
Explanation: Quality factor is approximately the number of oscillations required for energy to drop significantly. If energy halves in 50 cycles, $Q \approx 50$.
250. A damped oscillator has natural frequency $\omega_0 = 20 \, \text{rad/s}$ and damping coefficient $\beta = 10 \, \text{s}^{-1}$. What type of damping is this?
ⓐ. Underdamped
ⓑ. Critically damped
ⓒ. Overdamped
ⓓ. No damping
Correct Answer: Underdamped
Explanation: Critical damping occurs when $\beta = \omega_0$. Here, $\beta = 10$, but actually critical occurs when $ \beta = \omega_0$. Since $\beta < \omega_0$, this is **underdamped**.
251. The general equation of motion for a damped oscillator is
ⓐ. $m\ddot{x} + kx = 0$
ⓑ. $m\ddot{x} + b\dot{x} + kx = 0$
ⓒ. $\ddot{x} + \omega^2 x = 0$
ⓓ. $m\ddot{x} + mg = 0$
Correct Answer: $m\ddot{x} + b\dot{x} + kx = 0$
Explanation: The differential equation of a damped oscillator includes inertia, damping, and restoring terms. $m\ddot{x}$ is inertia, $b\dot{x}$ is damping force, and $kx$ is restoring force.
252. In the equation $m\ddot{x} + b\dot{x} + kx = 0$, the term $b\dot{x}$ represents
ⓐ. Restoring force
ⓑ. Damping force proportional to velocity
ⓒ. Inertial force
ⓓ. External force
Correct Answer: Damping force proportional to velocity
Explanation: $b\dot{x}$ is damping force, where $b$ is the damping coefficient. It always opposes motion and reduces amplitude exponentially with time.
253. The solution of the damped oscillator equation for underdamping is
ⓐ. $x(t) = A e^{-\beta t}\cos(\omega_d t + \phi)$
ⓑ. $x(t) = A \cos(\omega t + \phi)$
ⓒ. $x(t) = A e^{\beta t}\cos(\omega t)$
ⓓ. $x(t) = A\sin(\omega t)$
Correct Answer: $x(t) = A e^{-\beta t}\cos(\omega_d t + \phi)$
Explanation: For underdamping, the solution combines oscillatory cosine term with exponential decay. Here, $\beta = b/2m$ and $\omega_d = \sqrt{\omega_0^2 - \beta^2}$.
254. The damped angular frequency $\omega_d$ is given by
ⓐ. $\omega_d = \sqrt{\omega_0^2 - \beta^2}$
ⓑ. $\omega_d = \omega_0 + \beta$
ⓒ. $\omega_d = \omega_0^2 - \beta^2$
ⓓ. $\omega_d = \beta/\omega_0$
Correct Answer: $\omega_d = \sqrt{\omega_0^2 - \beta^2}$
Explanation: In damped oscillations, frequency decreases due to damping. $\omega_0 = \sqrt{k/m}$ is natural frequency, and $\beta = b/2m$ is damping constant.
255. Which condition leads to critical damping?
ⓐ. $\beta < \omega_0$
ⓑ. $\beta = \omega_0$
ⓒ. $\beta > \omega_0$
ⓓ. $\beta = 0$
Correct Answer: $\beta = \omega_0$
Explanation: Critical damping occurs when the damping constant equals natural frequency. The system returns to equilibrium quickly without oscillating.
256. If $\beta > \omega_0$, the system is said to be
ⓐ. Underdamped
ⓑ. Overdamped
ⓒ. Critically damped
ⓓ. Undamped
Correct Answer: Overdamped
Explanation: In overdamping, oscillations do not occur. The system returns slowly to equilibrium without overshooting.
257. For underdamped motion, the amplitude decreases with time as
ⓐ. $A(t) = A_0 \cos(\omega t)$
ⓑ. $A(t) = A_0 e^{-\beta t}$
ⓒ. $A(t) = A_0 \sin(\beta t)$
ⓓ. $A(t) = A_0/\beta t$
Correct Answer: $A(t) = A_0 e^{-\beta t}$
Explanation: The exponential factor $e^{-\beta t}$ describes amplitude decay. The rate of decay depends on the damping constant $\beta$.
258. Which of the following best represents overdamped motion?
ⓐ. Exponentially decaying oscillations
ⓑ. Non-oscillatory exponential decay
ⓒ. Oscillations of constant amplitude
ⓓ. Growth in amplitude with time
Correct Answer: Non-oscillatory exponential decay
Explanation: In overdamped motion, the system does not oscillate. Displacement decays exponentially to zero without crossing the equilibrium position.
259. A damped oscillator has $m = 1 \, \text{kg}, k = 100 \, \text{N/m}, b = 10 \, \text{Ns/m}$. Find the damping constant $\beta$.
ⓐ. 2.5 s$^{-1}$
ⓑ. 5 s$^{-1}$
ⓒ. 10 s$^{-1}$
ⓓ. 20 s$^{-1}$
Correct Answer: 5 s$^{-1}$
Explanation: $\beta = b/2m = 10/(2 \times 1) = 5 \, \text{s}^{-1}$.
260. For the same system (m = 1 kg, k = 100 N/m, b = 10 Ns/m), find natural frequency $\omega_0$.
ⓐ. 5 rad/s
ⓑ. 10 rad/s
ⓒ. 20 rad/s
ⓓ. 100 rad/s
Correct Answer: 10 rad/s
Explanation: Natural frequency $\omega_0 = \sqrt{k/m} = \sqrt{100/1} = 10 \, \text{rad/s}$.
261. For the system in Q259 and Q260, determine whether it is underdamped, overdamped, or critically damped.
ⓐ. Underdamped
ⓑ. Overdamped
ⓒ. Critically damped
ⓓ. Undamped
Correct Answer: Underdamped
Explanation: Since $\beta = 5 \, \text{s}^{-1}$ and $\omega_0 = 10 \, \text{rad/s}$, we have $\beta < \omega_0$. Thus, the system is underdamped and oscillates with decreasing amplitude.
262. What are forced oscillations?
ⓐ. Oscillations maintained only by restoring force
ⓑ. Oscillations that die out naturally due to damping
ⓒ. Oscillations maintained by an external periodic driving force
ⓓ. Oscillations that occur only in vacuum
Correct Answer: Oscillations maintained by an external periodic driving force
Explanation: Forced oscillations occur when an external periodic force drives the system. Even if natural oscillations decay due to damping, the system continues oscillating at the frequency of the external force.
263. Which of the following is the general equation of motion for a forced oscillator?
ⓐ. $m\ddot{x} + b\dot{x} + kx = F_0 \cos(\omega t)$
ⓑ. $m\ddot{x} + kx = 0$
ⓒ. $m\ddot{x} + b\dot{x} = 0$
ⓓ. $m\ddot{x} + kx = F_0$
Correct Answer: $m\ddot{x} + b\dot{x} + kx = F_0 \cos(\omega t)$
Explanation: The complete equation includes inertia, damping, restoring, and the external driving force $F_0 \cos(\omega t)$, which sustains oscillations.
264. In forced oscillations, the frequency of oscillation of the system equals
ⓐ. Natural frequency of the system
ⓑ. Driving frequency of the external force
ⓒ. Twice the natural frequency
ⓓ. Zero
Correct Answer: Driving frequency of the external force
Explanation: In forced oscillations, the system oscillates with the frequency of the driving force, not its natural frequency, though amplitude depends on how close driving frequency is to natural frequency.
265. Which of the following is a characteristic of forced oscillations?
ⓐ. Amplitude is always constant
ⓑ. Frequency is controlled by external force
ⓒ. No damping occurs
ⓓ. Oscillations stop immediately if external force is applied
Correct Answer: Frequency is controlled by external force
Explanation: Unlike free oscillations, where frequency depends on system parameters, in forced oscillations the external driving frequency dominates.
266. A child is pushed periodically on a swing to keep it moving. This is an example of
ⓐ. Free oscillation
ⓑ. Damped oscillation
ⓒ. Forced oscillation
ⓓ. Random motion
Correct Answer: Forced oscillation
Explanation: The swing loses energy due to air resistance and friction, but when the child is pushed periodically, the external force maintains the oscillations.
267. Which parameter determines the steady-state amplitude of forced oscillations?
ⓐ. Initial displacement
ⓑ. Driving frequency and damping
ⓒ. Mass of oscillator only
ⓓ. Natural frequency only
Correct Answer: Driving frequency and damping
Explanation: Steady-state amplitude depends on the relation between driving frequency, natural frequency, and damping coefficient. Near resonance, amplitude can become very large.
268. When a periodic force acts on a damped oscillator, what happens to the transient part of motion?
ⓐ. It grows indefinitely
ⓑ. It remains forever
ⓒ. It decays with time and only forced motion remains
ⓓ. It oscillates randomly
Correct Answer: It decays with time and only forced motion remains
Explanation: The complete solution is the sum of transient (natural) oscillations and steady-state (forced) oscillations. With damping, transient oscillations decay, leaving only forced oscillations.
269. In mechanical systems, forced oscillations are often produced by
ⓐ. Internal restoring forces
ⓑ. Frictional forces only
ⓒ. External periodic driving forces like engines or vibrations
ⓓ. Random thermal motion
Correct Answer: External periodic driving forces like engines or vibrations
Explanation: Examples include vibrations in machines driven by motors or rotating parts, where external periodic forces drive the oscillations.
270. Which of the following statements is true for forced oscillations?
ⓐ. System oscillates only at its natural frequency
ⓑ. System oscillates at the driving frequency regardless of natural frequency
ⓒ. Amplitude is maximum when driving frequency is far from natural frequency
ⓓ. Damping has no effect on amplitude
Correct Answer: System oscillates at the driving frequency regardless of natural frequency
Explanation: The frequency of forced oscillations always matches the frequency of the applied external periodic force, though amplitude depends on resonance conditions and damping.
271. Which type of oscillation best describes the operation of a radio receiver tuned to a station frequency?
ⓐ. Free oscillations
ⓑ. Damped oscillations
ⓒ. Forced oscillations
ⓓ. Random oscillations
Correct Answer: Forced oscillations
Explanation: In a radio receiver, the external electromagnetic waves act as driving forces, forcing the circuit to oscillate at the frequency of the broadcast signal.
272. What is resonance in oscillatory systems?
ⓐ. When damping force becomes maximum
ⓑ. When the driving frequency equals the natural frequency of the system
ⓒ. When amplitude is minimum due to high frequency
ⓓ. When oscillations die out immediately
Correct Answer: When the driving frequency equals the natural frequency of the system
Explanation: Resonance occurs when the frequency of the external driving force matches the system’s natural frequency, producing maximum amplitude oscillations.
273. The frequency at which resonance occurs is called
ⓐ. Damping frequency
ⓑ. Resonant frequency
ⓒ. Driving frequency
ⓓ. Zero frequency
Correct Answer: Resonant frequency
Explanation: Resonant frequency is the natural frequency of the system at which maximum amplitude is achieved under forced oscillations.
274. A tuning fork resonates when struck near another tuning fork of the same frequency. This happens because
ⓐ. Both are damped oscillators
ⓑ. Energy transfers efficiently at equal frequencies
ⓒ. The second fork generates its own oscillations
ⓓ. Air resistance drives oscillations
Correct Answer: Energy transfers efficiently at equal frequencies
Explanation: Resonance allows energy transfer between oscillators with matching natural frequencies, causing the second tuning fork to vibrate without being struck.
275. In forced oscillations, maximum amplitude occurs when
ⓐ. Driving frequency ≫ natural frequency
ⓑ. Driving frequency ≪ natural frequency
ⓒ. Driving frequency = natural frequency
ⓓ. Driving frequency = 0
Correct Answer: Driving frequency = natural frequency
Explanation: At resonance, the system responds most strongly to external driving force, producing maximum amplitude oscillations.
276. Which of the following everyday examples demonstrates resonance?
ⓐ. A child pushed at the right frequency on a swing
ⓑ. A car engine producing constant vibrations
ⓒ. A pendulum losing energy due to friction
ⓓ. A mass sliding on a rough surface
Correct Answer: A child pushed at the right frequency on a swing
Explanation: The swing oscillates with large amplitude when pushed at its natural frequency, a perfect example of resonance in daily life.
277. A mechanical system has natural frequency $\omega_0 = 20 \, \text{rad/s}$. At what driving frequency will resonance occur?
ⓐ. 10 rad/s
ⓑ. 15 rad/s
ⓒ. 20 rad/s
ⓓ. 25 rad/s
Correct Answer: 20 rad/s
Explanation: Resonance occurs when driving frequency matches the natural frequency, hence at $\omega = \omega_0 = 20 \, \text{rad/s}$.
278. Which condition reduces the sharpness of resonance?
ⓐ. High stiffness
ⓑ. Low damping
ⓒ. High damping
ⓓ. Large amplitude
Correct Answer: High damping
Explanation: With greater damping, amplitude does not rise sharply near resonance, making the resonance curve broad and less sharp.
279. A series RLC circuit resonates when
ⓐ. Resistance = Capacitance
ⓑ. Inductive reactance = Capacitive reactance
ⓒ. Resistance = Inductance
ⓓ. Inductance = Capacitance
Correct Answer: Inductive reactance = Capacitive reactance
Explanation: In RLC circuits, resonance occurs when $X_L = X_C$, i.e., $\omega L = \tfrac{1}{\omega C}$. The circuit impedance is minimum and current maximum at resonance.
280. Which physical quantity is maximum at resonance in mechanical systems?
ⓐ. Frequency
ⓑ. Energy dissipation
ⓒ. Amplitude of oscillation
ⓓ. Damping coefficient
Correct Answer: Amplitude of oscillation
Explanation: Resonance produces large amplitudes because energy input from the external force matches the system’s natural oscillatory requirements.
281. Why is resonance sometimes dangerous in engineering structures?
ⓐ. It decreases natural frequency
ⓑ. It reduces restoring force
ⓒ. It produces excessively large amplitudes that can cause failure
ⓓ. It eliminates damping completely
Correct Answer: It produces excessively large amplitudes that can cause failure
Explanation: Resonance can cause catastrophic failures in bridges, buildings, or machines if external periodic forces (like wind or earthquakes) match their natural frequency, leading to structural damage.
282. In an LCR circuit at resonance, the impedance becomes
ⓐ. Maximum
ⓑ. Minimum
ⓒ. Infinite
ⓓ. Zero always
Correct Answer: Minimum
Explanation: At resonance in an LCR series circuit, inductive reactance and capacitive reactance cancel each other ($X_L = X_C$). Hence, impedance reduces to $Z = R$, the minimum possible value.
283. At resonance, the current in an LCR series circuit is
ⓐ. Minimum
ⓑ. Maximum
ⓒ. Zero
ⓓ. Constant independent of frequency
Correct Answer: Maximum
Explanation: Since impedance is minimum at resonance, the current amplitude becomes maximum, given by $I = V/Z$.
284. Which of the following mechanical systems shows resonance phenomenon?
ⓐ. A pendulum being pushed periodically at its natural frequency
ⓑ. A car moving at constant speed
ⓒ. A mass sliding on a horizontal surface without oscillation
ⓓ. A falling object under gravity
Correct Answer: A pendulum being pushed periodically at its natural frequency
Explanation: When the periodic push matches pendulum’s natural frequency, resonance occurs and swing amplitude becomes maximum.
285. Which of the following is an application of resonance in acoustics?
ⓐ. A flute producing sound
ⓑ. Noise due to friction
ⓒ. A person clapping
ⓓ. A guitar string breaking
Correct Answer: A flute producing sound
Explanation: Air column inside a flute resonates at natural frequencies, amplifying certain harmonics and producing musical notes.
286. In mechanical vibrations, why are shock absorbers filled with oil?
ⓐ. To reduce restoring force
ⓑ. To increase damping and prevent resonance damage
ⓒ. To increase mass of the suspension
ⓓ. To make oscillations sharper
Correct Answer: To increase damping and prevent resonance damage
Explanation: Shock absorbers use viscous damping to dissipate vibrational energy, avoiding dangerous resonance effects in vehicles.
287. The condition for resonance in an RLC circuit is
ⓐ. $\omega L = \tfrac{1}{\omega C}$
ⓑ. $L = C$
ⓒ. $R = L$
ⓓ. $R = C$
Correct Answer: $\omega L = \tfrac{1}{\omega C}$
Explanation: Resonance occurs when inductive reactance equals capacitive reactance. At this point, circuit impedance is minimum and current is maximum.
288. Which of the following uses resonance in radio tuning?
ⓐ. Increasing current in resistors
ⓑ. Selecting signals of a specific frequency
ⓒ. Converting AC to DC
ⓓ. Reducing resistance of wire
Correct Answer: Selecting signals of a specific frequency
Explanation: Radio receivers use an LCR circuit tuned to resonate at the desired station frequency. Other frequencies are filtered out.
289. In acoustics, resonance is useful because it
ⓐ. Reduces amplitude of sound waves
ⓑ. Increases amplitude and loudness of specific frequencies
ⓒ. Stops oscillations
ⓓ. Eliminates harmonics
Correct Answer: Increases amplitude and loudness of specific frequencies
Explanation: Resonance amplifies natural frequencies of instruments, producing louder and richer sound. For example, resonance boxes in guitars and violins.
290. Which of the following is a destructive example of resonance in mechanical vibrations?
ⓐ. Earthquake waves matching the natural frequency of a building
ⓑ. A car engine working normally
ⓒ. A pendulum used in clocks
ⓓ. A child’s swing
Correct Answer: Earthquake waves matching the natural frequency of a building
Explanation: When seismic waves resonate with a building’s natural frequency, amplitudes grow dangerously, leading to collapse.
291. In electrical engineering, resonance is exploited in
ⓐ. Transformers
ⓑ. Oscillatory circuits for tuning and frequency selection
ⓒ. DC power supplies
ⓓ. Resistive heating
Correct Answer: Oscillatory circuits for tuning and frequency selection
Explanation: Resonance is used in LC circuits of radios, TVs, and communication systems to select desired frequency signals with high precision.
292. What distinguishes a nonlinear oscillatory system from a linear one?
ⓐ. Restoring force is always proportional to displacement
ⓑ. Restoring force depends on higher powers of displacement or velocity
ⓒ. Frequency is independent of amplitude
ⓓ. Oscillations always remain sinusoidal
Correct Answer: Restoring force depends on higher powers of displacement or velocity
Explanation: In linear oscillations (like SHM), restoring force $F = -kx$. In nonlinear systems, restoring force might be $F = -kx - \alpha x^3$, where $\alpha$ introduces nonlinearity. This causes amplitude-dependent frequency and non-sinusoidal motion. Such nonlinearities appear in stiff springs, pendulums at large amplitudes, and electrical circuits with nonlinear elements. Equations: $$ F = -kx \quad \text{(linear SHM)}$$ $$ F = -kx - \alpha x^3 \quad \text{(nonlinear oscillator)}$$ $$ m\ddot{x} + kx + \alpha x^3 = 0 \quad \text{(Duffing equation form)}$$ $$ \omega(A) \approx \omega_0 \left(1 + \tfrac{3}{8}\tfrac{\alpha A^2}{k}\right) \quad \text{(frequency depends on amplitude)}$$
293. A pendulum at large amplitudes is an example of a nonlinear oscillator because
ⓐ. $\sin\theta \approx \theta$ always holds
ⓑ. Restoring torque is $\tau = -mgL\sin\theta$, which is nonlinear in $\theta$
ⓒ. Damping is absent
ⓓ. Energy is not conserved
Correct Answer: Restoring torque is $\tau = -mgL\sin\theta$, which is nonlinear in $\theta$
Explanation: For small oscillations, $\sin\theta \approx \theta$, making motion linear. For large angles, higher-order terms in $\sin\theta$ cannot be ignored, producing nonlinear behavior where frequency depends on amplitude. Equations: $$ \tau = -mgL\sin\theta$$ $$ I\ddot{\theta} + mgL\sin\theta = 0$$ $$ \ddot{\theta} + \omega_0^2\theta + \tfrac{1}{6}\omega_0^2\theta^3 + \dots = 0$$ $$ T(A) = 2\pi\sqrt{\tfrac{L}{g}} \left[1 + \tfrac{1}{16}A^2 + \dots \right]$$
294. Which feature is characteristic of nonlinear oscillations?
ⓐ. Constant period regardless of amplitude
ⓑ. Period depends on amplitude of oscillation
ⓒ. Oscillations always decay exponentially
ⓓ. Oscillations stop after one cycle
Correct Answer: Period depends on amplitude of oscillation
Explanation: In linear SHM, period $T = 2\pi \sqrt{\tfrac{m}{k}}$, independent of amplitude. In nonlinear systems, period is modified by amplitude-dependent terms. This feature is crucial in real systems like large pendulums or nonlinear electrical circuits.
Equations: $$T_0 = 2\pi \sqrt{\tfrac{m}{k}} \quad \text{(linear case)}$$ $$T(A) = T_0 \left(1 + \tfrac{3}{8}\tfrac{\alpha A^2}{k} \right) \quad \text{(Duffing approximation)}$$ $$\omega(A) = \tfrac{2\pi}{T(A)}$$ $$\Delta T \propto A^2 \quad \text{(period increases with amplitude)}$$
295. A nonlinear restoring force in a spring system is given by $F = -kx - \alpha x^3$. Which effect does the cubic term introduce?
ⓐ. It makes oscillations purely sinusoidal
ⓑ. It introduces anharmonicity and amplitude dependence
ⓒ. It cancels damping
ⓓ. It eliminates resonance
Correct Answer: It introduces anharmonicity and amplitude dependence
Explanation: The cubic nonlinearity leads to the Duffing oscillator model. Such systems show stiffening ($\alpha > 0$) or softening ($\alpha < 0$) behavior, where frequency shifts with amplitude. These nonlinear effects are critical in engineering structures and micro-electro-mechanical systems (MEMS). Equations: $$m\ddot{x} + kx + \alpha x^3 = 0$$ $$\omega(A) \approx \omega_0 \left(1 + \tfrac{3}{8}\tfrac{\alpha A^2}{k} \right)$$ $$E = \tfrac{1}{2}kx^2 + \tfrac{1}{4}\alpha x^4$$ $$x(t) \approx A\cos(\omega(A) t) \quad \text{with frequency shift}$$
296. Which of the following is NOT true about nonlinear oscillations?
ⓐ. They can lead to chaotic motion under certain conditions
ⓑ. They always have sinusoidal solutions
ⓒ. Their frequency often depends on amplitude
ⓓ. They appear in pendulums, nonlinear springs, and electrical circuits
Correct Answer: They always have sinusoidal solutions
Explanation: Nonlinear oscillations rarely remain sinusoidal. Depending on system parameters, motion may include higher harmonics, amplitude dependence, or even chaotic trajectories. Linear oscillations are perfectly sinusoidal, but nonlinear ones are not. Equations: $$\ddot{x} + \omega_0^2 x + \alpha x^3 = 0 \quad \text{(Duffing oscillator)}$$ $$x(t) = A\cos(\omega t) + B\cos(3\omega t) + \dots$$ $$\omega(A) \propto f(A) \quad \text{(amplitude dependence)}$$ $$\text{Possible chaotic motion when driven: } m\ddot{x} + b\dot{x} + kx + \alpha x^3 = F\cos(\omega t)$$
297. Which effect shows that nonlinear oscillations differ from linear SHM?
ⓐ. Frequency remains fixed regardless of amplitude
ⓑ. Frequency changes with amplitude of oscillation
ⓒ. Restoring force is always linear in displacement
ⓓ. Energy is independent of amplitude
Correct Answer: Frequency changes with amplitude of oscillation
Explanation: In linear SHM, frequency $\omega_0 = \sqrt{k/m}$ is independent of amplitude. In nonlinear systems, higher-order terms like $-\alpha x^3$ in the restoring force make the oscillation frequency amplitude-dependent. For large amplitudes, this leads to stiffening ($\alpha > 0$) or softening ($\alpha < 0$) behavior. Equations: $$ F = -kx - \alpha x^3$$ $$ m\ddot{x} + kx + \alpha x^3 = 0$$ $$ \omega(A) \approx \omega_0 \left(1 + \tfrac{3}{8}\tfrac{\alpha A^2}{k}\right)$$ $$ T(A) = \tfrac{2\pi}{\omega(A)} \quad \Rightarrow \quad T \text{ depends on } A$$
298. A simple pendulum at large amplitude demonstrates nonlinear effects because
ⓐ. Restoring force is proportional to $\theta$
ⓑ. Restoring torque is $\tau = -mgL\sin\theta$, not linear in $\theta$
ⓒ. Time period is strictly constant for all amplitudes
ⓓ. Energy exchange vanishes at large amplitude
Correct Answer: Restoring torque is $\tau = -mgL\sin\theta$, not linear in $\theta$
Explanation: At small angles, $\sin\theta \approx \theta$, so the pendulum behaves linearly. For larger amplitudes, higher-order terms contribute, causing the period to increase with amplitude. This is a nonlinear effect seen in real pendulums. Equations: $$ \tau = -mgL\sin\theta$$ $$ \ddot{\theta} + \tfrac{g}{L}\sin\theta = 0$$ $$ T(A) = 2\pi \sqrt{\tfrac{L}{g}} \left( 1 + \tfrac{1}{16}A^2 + \tfrac{11}{3072}A^4 + \cdots \right)$$ $$ T \uparrow \text{ as amplitude } A \uparrow$$
299. Which nonlinear effect occurs in systems like stiff springs or MEMS devices with cubic nonlinearity?
ⓐ. Frequency decreases with amplitude (softening effect)
ⓑ. Frequency increases with amplitude (hardening effect)
ⓒ. Frequency stays constant
ⓓ. Oscillations stop immediately
Correct Answer: Frequency increases with amplitude (hardening effect)
Explanation: When restoring force includes a positive cubic term $+\alpha x^3$ with $\alpha > 0$, oscillations stiffen. This causes the oscillation frequency to increase with amplitude (hardening spring effect). Equations: $$ F = -kx - \alpha x^3, \quad \alpha > 0$$ $$ m\ddot{x} + kx + \alpha x^3 = 0$$ $$ \omega(A) \approx \omega_0 \left(1 + \tfrac{3}{8}\tfrac{\alpha A^2}{k}\right), \quad \omega \uparrow \text{ with } A$$ $$ \Delta\omega \propto A^2$$
300. A softening nonlinear oscillator ($\alpha < 0$) is characterized by
ⓐ. Frequency increases with amplitude
ⓑ. Frequency decreases with amplitude
ⓒ. Period remains constant
ⓓ. Oscillations always chaotic
Correct Answer: Frequency decreases with amplitude
Explanation: Negative cubic term weakens the restoring force at higher amplitudes, making oscillations slower. This softening effect is seen in rubber bands and flexible mechanical systems under large oscillations.
Equations: $$ F = -kx + |\alpha|x^3, \quad \alpha < 0$$ $$ \omega(A) \approx \omega_0 \left(1 - \tfrac{3}{8}\tfrac{|\alpha| A^2}{k}\right)$$ $$ T(A) \approx T_0 \left(1 + \tfrac{3}{8}\tfrac{|\alpha| A^2}{k}\right)$$ $$ \omega \downarrow \text{ as amplitude } A \uparrow$$
This section covers Class 11 Physics MCQs – Chapter 14: Oscillations (Part 3). Here, the focus shifts to advanced topics such as damped oscillations, forced oscillations, and resonance. These ideas are crucial not only for the NCERT/CBSE syllabus but also for high-level applications in board exams, JEE, NEET, and competitive tests. Students will explore how oscillations lose energy over time (damping), how external forces affect oscillatory systems (forced oscillations), and the practical significance of resonance in engineering and medical applications. The complete chapter offers 455 solved MCQs with step-by-step explanations, divided into 5 parts. This part contains the third set of 100 MCQs, designed to challenge your understanding and improve conceptual depth.
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