Q. No.Identify the correct definition:
1.
If after every certain interval of time, a particle repeats its motion, then the motion is called periodic motion.
2.
To and fro motion of a particle is called oscillatory motion.
3.
Oscillatory motion described in terms of single sine and cosine functions is called simple harmonic motion.
4.
All of the above
A particle of mass \(m\) and charge \(\text-q\) moves diametrically through a uniformly charged sphere of radius \(R\) with total charge \(Q\). The angular frequency of the particle's simple harmonic motion, if its amplitude \(<R\), is given by:
1. \(\sqrt{\dfrac{qQ}{4 \pi \varepsilon_0 ~mR} }\)
2. \(\sqrt{\dfrac{qQ}{4 \pi \varepsilon_0 ~mR^2} }\)
3. \(\sqrt{\dfrac{qQ}{4 \pi \varepsilon_0 ~mR^3}}\)
4. \( \sqrt{\dfrac{m}{4 \pi \varepsilon_0 ~qQ} }\)
A spring having a spring constant of \(1200\) N/m is mounted on a horizontal table as shown in the figure. A mass of \(3\) kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of \(2.0\) cm and released. The frequency of oscillations will be:
| 1. | \(3.0~\text{s}^{-1}\) | 2. | \(2.7~\text{s}^{-1}\) |
| 3. | \(1.2~\text{s}^{-1}\) | 4. | \(3.2~\text{s}^{-1}\) |
The figure shows the circular motion of a particle. The radius of the circle, the period, the sense of revolution, and the initial position are indicated in the figure. The simple harmonic motion of the \(\mathrm{x\text-}\)projection of the radius vector of the rotating particle \(P\) will be:
1. \(x \left( t \right) = B\) \(\text{sin} \left(\dfrac{2 πt}{30}\right)\)
2. \(x \left( t \right) = B\) \(\text{cos} \left(\dfrac{πt}{15}\right)\)
3. \(x \left( t \right) = B\) \(\text{sin} \left(\dfrac{πt}{15} + \dfrac{\pi}{2}\right)\)
4. \(x \left( t \right) = B\) \(\text{cos} \left(\dfrac{πt}{15} + \dfrac{\pi}{2}\right)\)
| 1. | Phase of the oscillator is the same at \(t = 0~\text{s}~\text{and}~t = 2~\text{s}\). |
| 2. | Phase of the oscillator is the same at \(t = 2~\text{s}~\text{and}~t = 6~\text{s}\). |
| 3. | Phase of the oscillator is the same at \(t = 1~\text{s}~\text{and}~t = 7~\text{s}\). |
| 4. | Phase of the oscillator is the same at \(t = 1~\text{s}~\text{and}~t = 5~\text{s}\). |
| 1. | \(1,2~\text{and}~4\) | 2. | \(1~\text{and}~3\) |
| 3. | \(2~\text{and}~4\) | 4. | \(3~\text{and}~4\) |
| 1. | \(e^{\omega t}\) | 2. | \(\text{log}_e(\omega t)\) |
| 3. | \(\text{sin}\omega t+ \text{cos}\omega t\) | 4. | \(e^{-\omega t}\) |
All the surfaces are smooth and the system, given below, is oscillating with an amplitude \(\mathrm{A}.\) What is the extension of spring having spring constant \(\mathrm{k_1},\) when the block is at the extreme position?
| 1. | \({k_1 \over k_1+k_2} \text{A}\) | 2. | \({k_2A \over k_1+k_2}\) |
| 3. | \(\mathrm{A}\) | 4. | \(\text{A} \over 2\) |
1. \(x= 10\sin\left(\pi t+\frac{\pi}{6}\right)\)
2. \(x= 10\sin\left(\pi t\right)\)
3. \(x= 10\cos\left(\pi t\right)\)
4. \(x= 5\sin\left(\pi t+\frac{\pi}{6}\right)\)
| 1. | \(2v_0 \over \sqrt{3}\) | 2. | \(\sqrt{2}v_0 \over 3\) |
| 3. | \({2 \over 3}v_0\) | 4. | \(\sqrt{\frac{2}{3}}v_0\) |
| 1. | \(0.01~\text{Hz}\) | 2. | \(0.02~\text{Hz}\) |
| 3. | \(0.03~\text{Hz}\) | 4. | \(0.04~\text{Hz}\) |
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