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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. |

Subtopic: Types of Motion |

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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. \( \sqrt{\dfrac{m}{4 \pi \varepsilon_0 ~qQ} }\)

Subtopic: Linear SHM |

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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 x-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. \(x \left( t \right) = B\) \(\text{cos} \left(\dfrac{πt}{15} + \dfrac{\pi}{2}\right)\)

Subtopic: Simple Harmonic Motion | Phasor Diagram |

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Displacement versus time curve for a particle executing SHM is shown in the figure. Choose the correct statement/s.

1. | Phase of the oscillator is the same at t =0 s and t = 2 s. |

2. | Phase of the oscillator is the same at t =2 s and t=6 s. |

3. | Phase of the oscillator is the same at t = 1 s and t=7 s. |

4. | Phase of the oscillator is the same at t=1 s and t=5 s. |

1. | 1, 2 and 4 | 2. | 1 and 3 |

3. | 2 and 4 | 4. | 3 and 4 |

Subtopic: Simple Harmonic Motion |

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A particle is executing SHM with time period T. If the time period of its total mechanical energy is T', then $\frac{{T}^{\text{'}}}{T}$ will be:

1. | 2 | 2. | \(1 \over 2\) |

3. | Zero | 4. | Infinite |

Subtopic: Energy of SHM |

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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 |

Subtopic: Types of Motion |

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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. | \(3.2~\text{s}^{-1}\) |

Subtopic: Spring mass system |

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Which of the following relationships between the acceleration '\(a\)' and the displacement '\(x\)' of a particle involves simple harmonic motion?

1. \(a = 0 . 7 x\)

2. \(a = - 200 x^{2} \)

3. \(a = - 10 x\)

4. \(a = 100 x^{3}\)

Subtopic: Simple Harmonic Motion |

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Which of the following examples represent simple harmonic motion?

1. | The rotation of the earth about its axis. |

2. | The motion of an oscillating mercury column in a U-tube. |

3. | General vibrations of a polyatomic molecule about its equilibrium position. |

4. | A fan rotating with a constant angular velocity. |

Subtopic: Simple Harmonic Motion |

62%

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From the given functions, identify the function which represents a periodic motion:

1. | \(e^{\omega t}\) | 2. | \(\text{log}_e(\omega t)\) |

3. | \(\text{sin}\omega t+ \text{cos}\omega t\) | 4. | \(e^{-\omega t}\) |

Subtopic: Types of Motion |

88%

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