Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
(a) Let us push the particle lightly against the spring C through displacement x.
As a result of this movement, the resultant force on the particle is kx.
The force on the particle due to springs A and B is \[\frac{kx}{\sqrt{2}}\] .
Total Resultant force \[= kx + \sqrt{\left( \frac{kx}{\sqrt{2}} \right)^2 + \left( \frac{kx}{\sqrt{2}} \right)^2}\]
= kx + kx = 2kx
Acceleration is given by \[= \frac{2kx}{m}\]
\[\text { Time period }= 2\pi\sqrt{\frac{\text { Displacement }}{\text { Acceleration }}}\]
\[ = 2\pi\sqrt{\frac{x}{2kx/m}}\]
\[ = 2\pi\sqrt{\frac{m}{2k}}\]
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 periodic motion?
An arrow released from a bow.
The length of the second’s pendulum in a clock is increased to 4 times its initial length. Calculate the number of oscillations completed by the new pendulum in one minute.
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 executes simple harmonic motion with a frequency v. The frequency with which the kinetic energy oscillates 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.
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.
A uniform plate of mass M stays horizontally and symmetrically on two wheels rotating in opposite direction in Figure . The separation between the wheels is L. The friction coefficient between each wheel and the plate is μ. Find the time period of oscillation of the plate if it is slightly displaced along its length and released.
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.
A uniform disc of radius r is to be suspended through a small hole made in the disc. Find the minimum possible time period of the disc for small oscillations. What should be the distance of the hole from the centre for it to have minimum time period?
Which of the following example represent periodic motion?
A hydrogen molecule rotating about its center of mass.
Which of the following example represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
The rotation of the earth about its axis.
Which of the following example represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
The motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lowermost point.
A simple pendulum of frequency n falls freely under gravity from a certain height from the ground level. Its frequency of oscillation.
Show that the motion of a particle represented by y = sin ωt – cos ωt is simple harmonic with a period of 2π/ω.
