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. | \(3~\text{and}~4\) |
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. \( \sqrt{\dfrac{m}{4 \pi \varepsilon_0 ~qQ} }\)
1. | \(e^{\omega t}\) | 2. | \(\text{log}_e(\omega t)\) |
3. | \(\text{sin}\omega t+ \text{cos}\omega t\) | 4. | \(e^{-\omega t}\) |
The rotation of the earth about its axis is:
1. | periodic motion. |
2. | simple harmonic motion. |
3. | periodic and simple harmonic motion. |
4. | non-periodic motion. |
1. | \(2\) | 2. | \(1 \over 2\) |
3. | Zero | 4. | Infinite |
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 |
1. | \(0.01~\text{Hz}\) | 2. | \(0.02~\text{Hz}\) |
3. | \(0.03~\text{Hz}\) | 4. | \(0.04~\text{Hz}\) |
A particle executing simple harmonic motion of amplitude \(5~\text{cm}\) has a maximum speed of \(31.4~\text{cm/s}.\) The frequency of its oscillation will be:
1. \(1~\text{Hz}\)
2. \(3~\text{Hz}\)
3. \(2~\text{Hz}\)
4. \(4~\text{Hz}\)