Advertisements
Advertisements
प्रश्न
Find the time period of mass M when displaced from its equilibrium position and then released for the system shown in figure.
Advertisements
उत्तर
For the calculation purpose. in this situation, we will neglect gravity because it is constant throughout and will not affect the net restoring force.
Let in the equilibrium position, the spring has extended by an amount x0.
Now, if the mass is given a further displacement downwards by an amount of x. The string and spring both should increase in length by x.
But. string is inextensible, hence the spring alone will contribute the total extension x + x = 2x, to lower the mass down by x from the initial equilibrium mean position x0. So, net extension in the spring (= 2x + x0)
Now force on the mass before bullying (in the x0 extension case)
F = 2T
But T = kx0 ......[Where k is spring constant]
∴ F = 2kx0 ......(i)
When the mass is lowered further by x,
F' = 2T'
But new spring length = (2x + x0)
∴ F' = 2k(2x + x0) ......(ii)
Restoring force on the system
`F_"restoring" = - [F^' - F]`
Using equations (i) and (ii), we get
`F_"restoring" = -[2k(2x + x_0) - 2kx_0]`
= `- [2 xx 2kx + 2kx_0 - 2kx_0]`
= `- 4kx`
or Ma = `- 4kx`
Where, a = acceleration .....(As, F = ma)
⇒ a = `- ((4k)/M)x`
k, M is constant.
∴ a ∝ – x
Hence, the motion is S.H.M
Comparing the above acceleration expression with standard SHM equation a = – ω2x, we get
`ω^2 = (4k)/M`
⇒ `ω = sqrt((4K)/M)`
∴ Time period T = `(2pi)/ω = (2pi)/sqrt((4K)/M) = 2pi sqrt(M/(4k))`
APPEARS IN
संबंधित प्रश्न
The acceleration due to gravity on the surface of moon is 1.7 ms–2. What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s? (g on the surface of earth is 9.8 ms–2)
Answer the following questions:
A time period of a particle in SHM depends on the force constant k and mass m of the particle: `T = 2pi sqrt(m/k)` A simple pendulum executes SHM approximately. Why then is the time
Answer the following questions:
What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?
A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?
The cylindrical piece of the cork of density of base area A and height h floats in a liquid of density `rho_1`. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
`T = 2pi sqrt((hrho)/(rho_1g)`
where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).
A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity v0 at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x0 and v0. [Hint: Start with the equation x = acos (ωt+θ) and note that the initial velocity is negative.]
A clock regulated by seconds pendulum, keeps correct time. During summer, length of pendulum increases to 1.005 m. How much will the clock gain or loose in one day?
(g = 9.8 m/s2 and π = 3.142)
Define practical simple pendulum
Show that, under certain conditions, simple pendulum performs the linear simple harmonic motion.
If the particle starts its motion from mean position, the phase difference between displacement and acceleration is ______.
If the maximum velocity and acceleration of a particle executing SHM are equal in magnitude, the time period will be ______.
The period of oscillation of a simple pendulum of constant length at the surface of the earth is T. Its time period inside mine will be ______.
A particle executing S.H.M. has a maximum speed of 30 cm/s and a maximum acceleration of 60 cm/s2. The period of oscillation is ______.
The length of a second’s pendulum on the surface of earth is 1 m. What will be the length of a second’s pendulum on the moon?
A body of mass m is situated in a potential field U(x) = U0 (1 – cos αx) when U0 and α are constants. Find the time period of small oscillations.
A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute S.H.M. with a time period. `T = 2πsqrt(m/(Apg))` where m is mass of the body and ρ is density of the liquid.
A particle at the end of a spring executes simple harmonic motion with a period t1, while the corresponding period for another spring is t2. If the period of oscillation with the two springs in series is T, then ______.
