Ans. a ∝ - x
i) Acceleration is directly proportional to the displacement.
ii) Acceleration is directed towards its mean position.
Q.2 Does frequency depends on amplitude for harmonic oscillators?
Ans. No. Frequency of harmonic oscillator is independent of amplitude. It depends upon time period T. f = 1 / T
Q.3 Can we realize an ideal simple pendulum?
Ans. No. Due to friction and weight of the string. For an ideal simple pendulum, the string should be massless, inextensible and suspended from frictionless support.
Q.4 What is the total distance traveled by an object moving with SHM in a time equal to its period,
if its amplitude is A?
Ans. T, is time for one complete vibration. Its maximum displacement, xo = r = A. so total distance traveled will be 4A.
Q.5 What happens to the period of a simple pendulum if its length is doubled? What happens if the suspended mass is doubled?
Ans. We have for simple pendulum, T = 2π√ l / g For l = 2l T = 2π√ 2 l / g = √2 x 2π√ l / g = √2 T So the time period increases by √2 (=1.414) times, as length is doubled. ii) There will be no change, when suspended mass is doubled. Since time period, T, is independent of mass, m.
Q.6 Does the acceleration of a simple harmonic oscillator remain constant during its motion? Is the acceleration ever zero? Explain.
Ans. No. Acceleration depends upon x, A = - ω2 x The acceleration is zero at mean position (x = 0) and it becomes maximum at extreme position (x = xo) so the acceleration of simple harmonic oscillator does not remain constant during its motion.
Q.7 What is meant by phase angle? Does it define angle between maximum displacement and the driving force?
Ans. i) Phase angle (or phase): “The angle θ = ωt which specifies the displacement as well as the direction of motion of the point executing SHM”. It indicates the state and direction of motion of a vibrating particle. ii) No. It does not define angle between maximum displacement and the driving force.
Q.8 Under what conditions does the addition of two simple harmonic motions produce a resultant, which is also simple harmonic?
Ans. Under the phenomenon of parallel superposition of same waves, and beats, two harmonic motions produce a resultant simple harmonic.
Q.9 Show that in SHM the acceleration is zero when the velocity is greatest and the velocity is zero when the acceleration is greatest.
Ans. We have for SHM; v = ω √ xo2 – x2 & a = - ω2 x At mean position, from the above equations, X = 0 then a = 0 & v = ω xo—maximum value, i.e. acceleration is zero and velocity is greatest. & at extreme positions; x = xo then v = 0 & a = -ω xo—maximum value. i. e. velocity is zero when acceleration is greatest.
Q.10 In relation to SHM, explain the equations;
(i) y = A sin (ω t + ϕ )
(ii) a = - ω2 x
Ans. i) y = A sin (ω t + ϕ ) initial phase
Inst. displ. Amplitude angle subtended in time t This equation shows that displacement of SHM as a function of amplitude and phase angle depending upon time. ii) a = - ω2 x where a = acceleration of a particle executing SHM ω = constant angular frequency x = instantaneous displacement from the mean position This equation shows that acceleration is directly proportional to displacement and is directed towards mean position.
Q.11 Explain the relation between total energy, potential energy and kinetic energy for a body oscillating with SHM.
Ans. For a body executing SHM; At mean position, x = 0 PE = ½ k x2 = ½ k (0)2 = 0 → minimum KE = ½ k xo2 (1 – x2 /xo2) = ½ k xo2 → maximum At extreme position, x = xo PE = ½ k x2 = ½ k xo2 → maximum & KE = ½ k xo2 (1 – x2 /xo2) = 0 → minimum At intermediate position, x = x Etotal = PE + KE = ½ k x2 + ½ k xo2 (1 – x2 /xo2) = ½ k xo2 We conclude that energy oscillate between maximum and minimum values and remain constant throughout equal to ½ k xo2 .
Q.12 Describe some common phenomena in which resonance plays an important role.
Ans. Important role of resonance: 1) Tuning radio/TV We change the frequency with knob. When it becomes equal to a particular transmitted station, resonance occurs. Then we receive amplified audio/video signals. 2) Microwave oven Microwaves (of frequency 2450 MHz) with λ = 12 cm, are absorbed due to resonance by water and fat molecules in the food, heating them up and so cooking the food. 3) Children’s swing In order to raise the swing to a great height, we must give it a push at the right moment and in the right direction. 4) Musical instruments In some instruments (e.g. drums) air columns resonate in the wooden box. In string instruments (e.g. sitar) strings resonate with their frequencies and loud music is heard.
Q.13 If a mass spring system is hung vertically and set into oscillations, why does the motion eventually stop?
Ans. Due to friction and air resistance mass-spring oscillating system eventually stops. When it oscillates, due to frictional forces energy is dissipated into heat and finally it stops.
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