Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given the potential energy associated with the field
U(x) = U0 (1 – cos αx) [∵ For conservative force f, we can write f = `(-du)/(dx)`] ......(i)
Now, Force F = `- (dU(x))/(dx)` .....[We have assumed the field to be conservative]
F = `- d/(dx) (U_0 - U_0 cos ax) = - U_0 a sin ax`
F = `- U_0 a^2x` [∵ For small oscillations ax is small, sin ax ≈ ax] ......(ii)
⇒ F ∝ (– x)
As, U0, a being constant.
∴ Motion is S.H.M for small oscillations.
The standard equation for S.H.M F = `- mω^2x` ......(iii)
Comparing equations (ii) and (iii), we get
`mω^2 = U_0a^2`
`ω^2 = (U_0a^2)/m` or `ω = sqrt((U_0a^2)/m)`
∴ Time period T = `(2pi)/ω = 2pi sqrt(m/(U_0a^2))`
APPEARS IN
संबंधित प्रश्न
The period of a conical pendulum in terms of its length (l), semi-vertical angle (θ) and acceleration due to gravity (g) is ______.
When the length of a simple pendulum is decreased by 20 cm, the period changes by 10%. Find the original length of the pendulum.
A spring having with a spring constant 1200 N m–1 is mounted on a horizontal table as shown in Fig. 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.
Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.
Answer the following questions:
The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger angles of oscillation, a more involved analysis shows that T is greater than `2pisqrt(1/g)` Think of a qualitative argument to appreciate this result.
Answer the following questions:
A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall?
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 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 motion of bob of the pendulum with small amplitude is linear S.H.M. Hence obtain an expression for its period. What are the factors on which its period depends?
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 ______.
When will the motion of a simple pendulum be simple harmonic?
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 tunnel is dug through the centre of the Earth. Show that a body of mass ‘m’ when dropped from rest from one end of the tunnel will execute simple harmonic motion.
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 ______.
