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
View Answer
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
View Answer
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): 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
View Answer
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
View Answer
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
View Answer
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 \(g\).
Assertion (A): In forced oscillations, the steady state motion of the particle (after natural oscillations die out) is SHM whose frequency is the frequency of the driving frequency \(\omega_d\), not the natural frequency \(\omega\) of the particle.
Reason (R): In forced oscillation \(\omega_d\) should be greater than natural frequency \(\omega\) of the 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
View Answer
In forced oscillations, the system eventually settles into oscillating at the frequency of the driving force, \(\omega_d\). So Assertion (A) is true. The driving frequency \(\omega_d\) can be any value (less than, equal to, or greater than) compared to the natural frequency \(\omega\). So Reason (R) is false.
Assertion (A): For a physical pendulum if distance of point of suspension from centre of mass increases time period first decreases then increases.
Reason (R): For a physical pendulum there is some distance from centre of mass at which frequency of oscillation is maximum.
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
View Answer
The time period of a physical pendulum is \(T = 2pi sqrt{(I_{CM} + mL^2)/(mgL)}\). Analyzing this function, \(T\) has a minimum value at \(L = sqrt{I_{CM}/m}\). This minimum time period corresponds to a maximum frequency. Thus, as \(L\) increases, \(T\) first decreases to a minimum and then increases. Both Assertion (A) and Reason (R) are true, and R correctly explains A.
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
View Answer
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
View Answer
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
View Answer
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.