Oscillations practice · Medium

Option B ($2:5$) is correct because the ratio is found by equating the kinetic energy to 84% of the total energy and solving for the displacement to amplitude ratio.

Option B (9N) is correct because the force in SHM is directly proportional to the displacement.

The correct answer is A ($8 \mathrm{~Nm}^{-1}, 0.628 \mathrm{~s}$) because the effective spring constant is the sum of individual spring constants and the time period is calculated using this effective constant.

Option D ($2 \pi \sqrt{2}$ seconds) is correct because the time period of a spring is calculated using the formula $T=2 \pi \sqrt{\frac{m}{k}}$.

Option A (If both Statement- I and Statement- II are true, and Statement - II is the correct explanation of Statement-I) is correct because the period of SHM is independent of the amplitude.

Option A ($\left[ {{{\left( {\frac{{{T_M}}}{T}} \right)}^2} - 1} \right]\frac{A}{{Mg}}$) is correct because it correctly relates the change in time period to Young's modulus and the added mass.

Option B (6) is correct because the minimum time taken to go from P to Q is calculated using the SHM equation and the given displacements.

Option D (5) is correct because the resultant amplitude is found using the formula for the superposition of two waves with a phase difference of 90 degrees.

The measurement of mass taken by a spring weighing balance is corrected at the place where the acceleration due to gravity is same with the place where the spring balance is calibrated for is correct because the reading depends on the local gravitational acceleration.

Option D ($\frac{(13 \hat{i}-29 \hat{j}-31 \hat{k})}{146}$) is correct because it is calculated using the formula for angular velocity, which is the cross product of the position vector and velocity vector, divided by the square of the magnitude of the position vector.

Option C is correct because the acceleration due to gravity decreases as you go deeper into the mine, which increases the period of oscillation of a simple pendulum.

The path of the particle is an ellipse (Option B) because the x and y coordinates satisfy the equation of an ellipse after integration and elimination of the time variable.

Option B (its time period is infinite at that point) is correct because the time period becomes infinite when the distance from the center of gravity is zero.

Option B (Restoring force) is correct because oscillatory motion requires a force that pulls the object back towards its equilibrium position.

Option B (10) is correct because the two pendulums will be in phase again after the longer pendulum completes 10 oscillations, as calculated by equating the time taken for a certain number of oscillations.

Option D (432 s) is correct because the time lost or gained is directly proportional to the percentage change in the length of the pendulum.

Option A is correct because the acceleration in SHM is greatest when the spring is at maximum displacement (fully compressed or extended) due to Hooke's Law.

Option C (1.6 kg) is correct because the mass is calculated using the period of oscillation formula for a spring-mass system.

Option A (${\pi ^2}m{a^2}{\mathrm{v}^2}$) is correct because the average kinetic energy is calculated as $\frac{1}{4}m a^2 \omega^2$, where $\omega = 2 \pi v$.

Option A (S-I & S-II are true) is correct because oscillatory motions are periodic, but not all periodic motions are oscillatory, and a simple pendulum exemplifies oscillatory motion.

Option C is correct because the oscillating particle and the driving force can have different phases, but this phase difference depends on the driving frequency relative to the natural frequency, not the initial conditions.

Option A (The distance of particle from its mean position) is correct because displacement in oscillatory motion is defined as the distance from the mean position.

Option A ($\mathrm{\pi }\,\,\mathrm{rad}$) is correct because acceleration in SHM is always opposite in direction to the displacement.

Option B (Its kinetic energy decreases while potential energy increases) is correct because as the swing rises, its speed decreases (decreasing KE) and its height increases (increasing PE).

Option C (Driving frequency is the same as the natural frequency of the oscillator) is correct because resonance occurs when the driving frequency matches the natural frequency.

Option D (Same at all position) is correct because in simple harmonic motion, the total energy remains constant, interchanging between kinetic and potential forms.

Option A (S-I & S-II are true) is correct because both statements accurately describe the behavior of damped oscillations.

Option B ($\infty$) is correct because in a freely falling cabin, the effective acceleration due to gravity is zero, leading to an infinite time period for the pendulum.

Option C (an ellipse) is correct because the relationship between velocity and displacement in SHM forms an ellipse equation.

Option B (If both assertion and reason are true but reason is not the correct explanation of assertion) is correct because the Earth's revolution is periodic but not oscillatory, and the reason correctly states the relationship between oscillatory and periodic motion, but it doesn't explain why the Earth's motion is periodic.