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Simple Harmonic Motion can be defined in two equivalent ways. One by the displacement equation x(t) = A cos(ωt + φ). What is the other equivalent way?

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Explanation

The context states: 'To summarise the discussion so far, simple harmonic motion can be defined in two equivalent ways, either by Eq. (13.4) for displacement or by Eq. (13.13) that gives its force law.' Eq. (13.13) is F(t) = -kx(t). (NCERT, page 267)

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To derive the displacement equation x(t) from the force law F(t) = -kx(t), one needs to perform:

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Explanation

The context mentions: 'Going from Eq. (13.4) to Eq. (13.13) required us to differentiate two times. Likewise, by integrating the force law Eq. (13.13) two times, we can get back Eq. (13.4).' (NCERT, page 267)

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For a spring-mass system executing SHM, if the mass 'm' is increased while the spring constant 'k' remains the same, the angular frequency ω will:

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Explanation

From the formula ω = √(k/m), given in the text (Eq. 13.14b), if 'm' increases and 'k' remains constant, ω will decrease. (NCERT, page 267)

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A body of mass 'm' is attached to two identical springs of spring constant 'k' each, connected in parallel with the mass between fixed supports. If the mass is displaced, the net restoring force F will be:

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Explanation

In Example 13.6, when two identical springs (spring constant k) are attached to a mass 'm' with fixed supports such that one spring elongates and the other compresses by 'x', the forces are F₁ = -kx and F₂ = -kx. The net force F = F₁ + F₂ = -2kx. (NCERT, page 268)

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What is the period of oscillation for a mass 'm' attached to two identical springs of spring constant 'k' each, connected as described in Example 13.6?

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Explanation

In Example 13.6, the net restoring force is F = -2kx. Comparing this with F = -K_eff x, we find the effective spring constant K_eff = 2k. The period of oscillation T = 2π√(m/K_eff). Substituting K_eff = 2k gives T = 2π√(m/2k). (NCERT, page 268)

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The total mechanical energy of a harmonic oscillator is given by E = ½kA². This energy:

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Explanation

The context states: 'The total mechanical energy of a harmonic oscillator is thus independent of time as expected for motion under any conservative force.' (NCERT, page 269)

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In SHM, at which state is the potential energy maximum and kinetic energy zero?

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Explanation

The text clarifies: 'For x = 0, the energy is kinetic; at the extremes x = ± A, it is all potential energy.' At extreme positions, the velocity is zero, hence kinetic energy is zero, and the potential energy is maximum. (NCERT, page 269)

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What is the relationship between angular frequency (ω), spring constant (k), and mass (m) for a particle in SHM?

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Explanation

The text clearly states the relationship as ω = k/m (13.14b). This is a crucial formula for SHM. (NCERT, page 267)

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According to the force law for SHM, if the displacement 'x' is positive, the restoring force 'F' will be:

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Explanation

The force law is F(t) = -kx(t). If x is positive, then F = -k(positive value), which results in a negative force. This indicates the force acts in the opposite direction to the displacement, towards the mean position. (NCERT, page 267-268)

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Which of the following describes the nature of bones in Aves enabling flight?

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Explanation

According to the NCERT text, 'Endoskeleton is fully ossified (bony) and the long bones are hollow with air cavities (pneumatic).' This adaptation reduces the weight of the bird, which is crucial for flight.

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