Advertisements
Advertisements
प्रश्न
Show that the motion of a particle represented by y = sin ωt – cos ωt is simple harmonic with a period of 2π/ω.
Advertisements
उत्तर
The given equation is in the form of a combination of two harmonic functions. We can write this equation in the form of a single harmonic (sine or cosine) function.
We have displacement function: `y = sin ωt - cos ωt`
`y = sqrt(2) (1/sqrt(2) * sin ωt - 1/sqrt(2) * cos ωt)`
= `sqrt(2)[cos (pi/4) * sin ωt - sin(pi/4) * cos ωt]`
= `sqrt(2) [sin (ωt - pi/4) = sqrt(2) [sin (ωt - pi/4)]]`
⇒ `y = sqrt(2) sin (ωt - pi/4)`
Comparing with the standard equation of S.H.M
`y = a sin(ωt + phi)` we get angular frequency of S.H.M, ω = `(2pi)/T` ⇒ `T = (2pi)/ω`
Hence the function represents S.H.M with a period T = 2π/ω.
APPEARS IN
संबंधित प्रश्न
The periodic time of a linear harmonic oscillator is 2π second, with maximum displacement of 1 cm. If the particle starts from extreme position, find the displacement of the particle after π/3 seconds.
Which of the following example represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
General vibrations of a polyatomic molecule about its equilibrium position.
A person goes to bed at sharp 10.00 pm every day. Is it an example of periodic motion? If yes, what is the time period? If no, why?
A particle moves in a circular path with a uniform speed. Its motion is
A particle is fastened at the end of a string and is whirled in a vertical circle with the other end of the string being fixed. The motion of the particle is
Consider a simple harmonic motion of time period T. Calculate the time taken for the displacement to change value from half the amplitude to the amplitude.
A spring stores 5 J of energy when stretched by 25 cm. It is kept vertical with the lower end fixed. A block fastened to its other end is made to undergo small oscillations. If the block makes 5 oscillations each second what is the mass of the block?
A particle of mass m is attatched to three springs A, B and C of equal force constants kas shown in figure . If the particle is pushed slightly against the spring C and released, find the time period of oscillation.
The left block in figure moves at a speed v towards the right block placed in equilibrium. All collisions to take place are elastic and the surfaces are frictionless. Show that the motions of the two blocks are periodic. Find the time period of these periodic motions. Neglect the widths of the blocks.
The ear-ring of a lady shown in figure has a 3 cm long light suspension wire. (a) Find the time period of small oscillations if the lady is standing on the ground. (b) The lady now sits in a merry-go-round moving at 4 m/s1 in a circle of radius 2 m. Find the time period of small oscillations of the ear-ring.
Find the time period of small oscillations of the following systems. (a) A metre stick suspended through the 20 cm mark. (b) A ring of mass m and radius r suspended through a point on its periphery. (c) A uniform square plate of edge a suspended through a corner. (d) A uniform disc of mass m and radius r suspended through a point r/2 away from the centre.
Find the number of oscillations performed per minute by a magnet is vibrating in the plane of a uniform field of 1.6 × 10-5 Wb/m2. The magnet has a moment of inertia 3 × 10-6 kg/m2 and magnetic moment 3 A m2.
The maximum speed of a particle executing S.H.M. is 10 m/s and maximum acceleration is 31.4 m/s2. Its periodic time is ______
Which of the following example represent periodic motion?
A freely suspended bar magnet displaced from its N-S direction and released.
A simple pendulum of frequency n falls freely under gravity from a certain height from the ground level. Its frequency of oscillation.
The displacement time graph of a particle executing S.H.M. is shown in figure. Which of the following statement is/are true?
- The force is zero at `t = (T)/4`.
- The acceleration is maximum at `t = (4T)/4`.
- The velocity is maximum at `t = T/4`.
- The P.E. is equal to K.E. of oscillation at `t = T/2`.
A person normally weighing 50 kg stands on a massless platform which oscillates up and down harmonically at a frequency of 2.0 s–1 and an amplitude 5.0 cm. A weighing machine on the platform gives the persons weight against time.
- Will there be any change in weight of the body, during the oscillation?
- If answer to part (a) is yes, what will be the maximum and minimum reading in the machine and at which position?
