Advertisements
Advertisements
Question
What is the ratio of maxmimum acceleration to the maximum velocity of a simple harmonic oscillator?
Advertisements
Solution
Let x = A sin ω is the displacement function of SHM.
Velocity, `v = (dx)/(dt) = Aω cosωt`
`v_max = Aω |cos ωt|_"max"`
= `Aω xx 1`
= `ωA` [∵ |cos ωt|max = 1] ......(i)
Acceleration, `a = (dv)/(dt) = - ωA * ωsinωt`
= `- ω^2A sinωt`
`|a_max| = |(- ω^2A)(+1)|` ......[∵ (sin ωt)max = 1]
`|a_max| = ω^2A` .......(ii)
From equations (i) and (ii), we get
`v_max/a_max = (ωA)/(ω^2A) = 1/ω`
⇒ `a_max/v_max = ω`
APPEARS IN
RELATED QUESTIONS
The average displacement over a period of S.H.M. is ______.
(A = amplitude of S.H.M.)
In a damped harmonic oscillator, periodic oscillations have _______ amplitude.
(A) gradually increasing
(B) suddenly increasing
(C) suddenly decreasing
(D) gradually decreasing
Hence obtain the expression for acceleration, velocity and displacement of a particle performing linear S.H.M.
A particle executes simple harmonic motion Let P be a point near the mean position and Q be a point near an extreme. The speed of the particle at P is larger than the speed at Q. Still the particle crosses Pand Q equal number of times in a given time interval. Does it make you unhappy?
The energy of system in simple harmonic motion is given by \[E = \frac{1}{2}m \omega^2 A^2 .\] Which of the following two statements is more appropriate?
(A) The energy is increased because the amplitude is increased.
(B) The amplitude is increased because the energy is increased.
Can a pendulum clock be used in an earth-satellite?
A block of known mass is suspended from a fixed support through a light spring. Can you find the time period of vertical oscillation only by measuring the extension of the spring when the block is in equilibrium?
The time period of a particle in simple harmonic motion is equal to the time between consecutive appearances of the particle at a particular point in its motion. This point is
The time period of a particle in simple harmonic motion is equal to the smallest time between the particle acquiring a particular velocity \[\vec{v}\] . The value of v is
A wall clock uses a vertical spring-mass system to measure the time. Each time the mass reaches an extreme position, the clock advances by a second. The clock gives correct time at the equator. If the clock is taken to the poles it will
Which of the following quantities are always zero in a simple harmonic motion?
(a) \[\vec{F} \times \vec{a} .\]
(b) \[\vec{v} \times \vec{r} .\]
(c) \[\vec{a} \times \vec{r} .\]
(d) \[\vec{F} \times \vec{r} .\]
Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel. The motion of the particle as seen from the earth is
(a) simple harmonic
(b) parabolic
(c) on a straight line
(d) periodic
A simple pendulum of length l is suspended through the ceiling of an elevator. Find the time period of small oscillations if the elevator (a) is going up with and acceleration a0(b) is going down with an acceleration a0 and (c) is moving with a uniform velocity.
A particle is subjected to two simple harmonic motions of same time period in the same direction. The amplitude of the first motion is 3.0 cm and that of the second is 4.0 cm. Find the resultant amplitude if the phase difference between the motions is (a) 0°, (b) 60°, (c) 90°.
In a simple harmonic oscillation, the acceleration against displacement for one complete oscillation will be __________.
Define the time period of simple harmonic motion.
Write short notes on two springs connected in series.
A container consist of hemispherical shell of radius 'r ' and cylindrical shell of height 'h' radius of same material and thickness. The maximum value h/r so that container remain stable equilibrium in the position shown (neglect friction) is ______.
