Assertion (A): General vibrations of a polyatomic molecule about its equilibrium position is periodic but not SHM.
Reason (R): A periodic motion can always be expressed as a sum of infinite number of harmonic motion with appropriated amplitude.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true. Complex vibrations of polyatomic molecules are periodic but generally not simple harmonic motion (SHM). Reason (R) is true. This is the principle of Fourier analysis, stating that any periodic motion can be decomposed into a sum of simple harmonic components. Reason (R) correctly explains why complex periodic motions (like polyatomic vibrations) are not SHM but can still be described as periodic.
Assertion (A): In SHM acceleration leads displacement by phase \(\pi\).
Reason (R): In SHM velocity leads displacement by phase \(\pi/2\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true. If displacement \(x = Asin(\omega t)\), then acceleration \(a = -A\omega^2sin(\omega t) = A\omega^2sin(\omega t + \pi)\). Reason (R) is true. Velocity \(v = A\omega cos(\omega t) = A\omega sin(\omega t + \pi/2)\). Both statements are true, but the phase relationship of velocity with displacement does not explain the phase relationship of acceleration with displacement directly; they are separate facts of SHM.
Assertion (A): Amplitude of SHM \(x = 4sin^2\omega t + 2cos^2\omega t + 2sin\omega t cos\omega t\) is \(sqrt{2}\).
Reason (R): Angular frequency of given equation is \(2\omega\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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The expression \(x = 4sin^2\omega t + 2cos^2\omega t + 2sin\omega t cos\omega t\) simplifies to \(x = 3 + sin(2\omega t) - cos(2\omega t)\). The oscillatory part is \(sin(2\omega t) - cos(2\omega t)\). Assertion (A) is true, its amplitude is \(\sqrt{1^2 + (-1)^2} = \sqrt{2}\). Reason (R) is true, the angular frequency is \(2\omega\). However, the angular frequency does not explain the specific amplitude value.
Assertion (A): For a physical pendulum period of oscillation is maximum about an axis passes through centre of mass.
Reason (R): A physical pendulum is in neutral equilibrium about centre of mass.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is false. If a physical pendulum is pivoted at its center of mass, it will be in neutral equilibrium and will not oscillate, so there is no period. Reason (R) is true. A body pivoted at its center of mass is indeed in neutral equilibrium. Thus, (A) is false and (R) is true.
Assertion (A): A spring block watch gives the correct time in orbiting satellite.
Reason (R): Time period of a spring block watch is independent of \(g\) and depends only on spring factor and mass of the block.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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The time period of a spring-mass system is \(T = 2\pi \sqrt{m/k}\). This equation shows that the time period is independent of the acceleration due to gravity \(g\). Therefore, a spring block watch would function correctly in an orbiting satellite where effective \(g\) is zero. Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
Assertion (A): If a pendulum clock is taken to a mountain top, its time period decreases.
Reason (R): Value of acceleration due to gravity is more at heights.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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The time period of a simple pendulum is \(T = 2\pi \sqrt{L/g}\). At a mountain top, the altitude increases, causing the acceleration due to gravity \(g\) to decrease. A decrease in \(g\) leads to an increase in \(T\). Thus, Assertion (A) is false. Reason (R) is also false as \(g\) decreases, not increases, at higher altitudes.
Assertion (A): In simple harmonic motion total mechanical energy can be negative also.
Reason (R): Potential energy is always negative and if it is greater than kinetic energy total mechanical energy will be negative.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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In simple harmonic motion, the total mechanical energy is \(E = frac{1}{2} kA^2\). Since the spring constant \(k\) and amplitude \(A\) are real and positive, the total energy \(E\) is always positive. Potential energy in SHM, \(U = frac{1}{2} kx^2\), is always positive or zero. Thus, both Assertion (A) and Reason (R) are false.
Assertion (A): Two particles are in SHM with same time period, same amplitude, same position and same speed are in the same phase.
Reason (R): Phase of particle depends on position and speed of particle.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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The position of a particle in SHM is given by \(x = A cos(omega t + phi)\) and velocity by \(v = -Aomega sin(omega t + phi)\). Given same amplitude \(A\), time period \(T\) (thus \(omega\)), position \(x\), and speed \(v\), the phase \(phi\) must be the same. Thus, A and R are true and R is the correct explanation of A.
Assertion (A): In damped oscillation both amplitude and frequency change with time.
Reason (R): Both amplitude and frequency vary exponentially.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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In a damped oscillation, the amplitude decreases exponentially with time, but the frequency of oscillation (for underdamped case) remains constant. Therefore, both Assertion (A) and Reason (R) are false.
Assertion (A): Time period of partially immersed spring block system is less than full immersed spring block system.
Reason (R): Time period of spring system is independent of changing values of g.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For a partially immersed block, the effective spring constant is \(k_{eff} = k + \rho_l A g\), leading to \(T_p = 2\pi \sqrt{m/(k + \rho_l A g)}\). For a fully immersed block, \(k_{eff} = k\), so \(T_f = 2\pi \sqrt{m/k}\). Since \(k + \rho_l A g > k\), \(T_p < T_f\). So A is true. The time period of a simple spring-mass system \(T = 2\pi \sqrt{m/k}\) is independent of \(g\). So R is true. However, R does not explain A because the change in period is due to effective spring constant, not general independence from \(g\).