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Q1. What are coupled oscillations?

Solution

A system of two bodies connected by a spring so that both can oscillate in a straight line along the length of the string is known as a coupled oscillator.  The oscillations produced by a coupled oscillator are called coupled oscillations.

Q2. Why are soldiers ordered to move out of step while crossing a bridge?

Solution

Every oscillator has its own frequency of oscillation.  When frequency of external force becomes equal to natural frequency of oscillations, resonance takes place and amplitude of vibration becomes maximum.  Bridge can be considered as stretched string.  If soldiers march in steps and frequency of marching steps is the same as that of frequency of bridge, then bridge will be set into resonant vibrations and hence collapse.  That is why soldiers are asked to break their steps.

Q3. How earthquakes some times cause disaster?

Solution

When the waves produced during earthquake have same frequency as that of the natural frequency of buildings, the resonance takes place and the building start vibrating with large amplitude and the amplitude is to large that the buildings collapse.  In this way the earthquakes causes big disaster.

Q4. An aeroplane passing over the building sometime causes rattling of windows of the building.  Explain why?

Solution

If the frequency of the sound produced by the engine of aeroplane passing over building is equal to the natural frequency of windows of the building, then resonance takes place and causes the rattling of window.

Q5. Write four characteristics of SHM.

Solution

(i)  In SHM, acceleration of the particle is directly proportional to its displacement, and directed towards the mean position. (ii)  It can be represented by a single harmonic function of sine or cosine. (iii)  Total energy of the particle executing SHM remains conserved. (iv)  It is a periodic motion.

Q6. At a certain speed of a bus, the body of the bus starts vibrating strongly.  Why?

Solution

At certain speed of bus, the frequency of the bus engine becomes equal to the natural frequency of its frame and hence resonant vibration of frame takes place, therefore bus begins to vibrate strongly.

Q7. Define the following : (i) damped oscillations (ii)  maintained oscillations (iii)  forced oscillations (iv)  resonant or sympathetic vibrations.

Solution

(i)  The oscillations in which the amplitude decreases progressively with the time are called damped oscillations. (ii)  When we feed energy back to the oscillations at the same rate at which it is dissipated, then the amplitude of such oscillations would remain constant with the time.  These oscillations are called maintained or sustained oscillations. (iii)  When an external periodic agent of frequency (n) is applied to an oscillator of natural frequency (n_o), the external agent is called the driver and the oscillating body is called the driven.  The driven oscillator ultimately settles down to the frequency of the driver.  Such oscillations that are forced upon the oscillator by the external periodic agent are known as the forced oscillations. (iv)  When the frequency of the driver (n) approaches the frequency of the driven (n_o), then the amplitude of the forced oscillations (and hence power drawn) becomes quite large.  The driver and the driven are said to be in resonance.  The phenomenon of setting a body into vibrations with its natural frequency by another body vibrating with the same frequency is called resonance.  

Q8. What will be the period of oscillation of a simple pendulum of length 100 cm in a spaceship in a geostationary orbit?

Solution

In any satellite orbiting the Earth (in any orbit), the condition of weightlessness exists (i.e. effective g = 0). Hence, the pendulum does not oscillate and its period is therefore infinity.

Q9. Two identical springs of force constant k each are connected in series.  What will be the spring factor of the combination when they are connected in (i)  Series (ii)  Parallel

Solution

(i) Series combination:  (ii) Parallel combination:

Q10. Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion? (i)  The rotation of the earth about its axis. (ii)  The motion of an oscillatory mercury column in a U-tube. (iii)  Motion of a ball-bearing inside a smooth curved bowl, when released from a point slightly above the lowermost position. (iv)  General vibration of a polyatomic molecule about its equilibrium configuration.

Solution

(ii) and (iii)  represent SHM as in both the cases the motion is to and fro about its mean position and the restoring force is proportional to displacement, but is in opposite direction to the displacement. (i) and (iv) represent periodic motion as in (i) there is no to and fro motion, and in (iv) the polyatomic molecule has a number of natural frequencies, and in general its vibration is a superposition of SHM's of a number of different frequencies.  This superposition is periodic but not SHM.