Advertisements
Advertisements
प्रश्न
State the differential equation of linear simple harmonic motion.
Advertisements
उत्तर
State the differential equation of linear S.H.M.
When a particle performs linear SHM. the force acting on the particle is always directed towards the mean position. The magnitude of the force is directly proportional to the magnitude of the displacement of the particle from the mean position. Thus, if `vecF` is the force acting on the particle when its displacement from the mean position is `vecx`
∴ `vecF = -kvecx` ....(1)
where the constant k, the force per unit displacement, is called the force constant. The minus sign indicates that the force and the displacement are oppositely directed.
The velocity of the particle is `(dvecx)/(dt)` and its acceleration is `(d^2vecx)/(dt^2)`
Let m be the mass of the particle
Force = mass x acceleration
∴ vecF = m `(d^2vecx)/(dt^2)`
Hence from Eq. (1), m `(d^2vecx)/(dt^2) = -k vecx`
`:. (d^2vecx)/(dt^2) + k/m vecx = 0` ....(2)
This is the differential equation of linear S.H.M.
APPEARS IN
संबंधित प्रश्न
Define phase of S.H.M.
Which of the following relationships between the acceleration a and the displacement x of a particle involve simple harmonic motion?
(a) a = 0.7x
(b) a = –200x2
(c) a = –10x
(d) a = 100x3
A particle executes S.H.M. with a period of 10 seconds. Find the time in which its potential energy will be half of its total energy.
Hence obtain the expression for acceleration, velocity and displacement of a particle performing linear S.H.M.
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.
A platoon of soldiers marches on a road in steps according to the sound of a marching band. The band is stopped and the soldiers are ordered to break the steps while crossing a bridge. Why?
A pendulum clock that keeps correct time on the earth is taken to the moon. It will run
Which of the following quantities are always negative in a simple harmonic motion?
(a) \[\vec{F} . \vec{a} .\]
(b) \[\vec{v} . \vec{r} .\]
(c) \[\vec{a} . \vec{r} .\]
(d)\[\vec{F} . \vec{r} .\]
Which of the following quantities are always positive in a simple harmonic motion?
A particle moves in the X-Y plane according to the equation \[\overrightarrow{r} = \left( \overrightarrow{i} + 2 \overrightarrow{j} \right)A\cos\omega t .\]
The motion of the particle is
(a) on a straight line
(b) on an ellipse
(c) periodic
(d) simple harmonic
An object is released from rest. The time it takes to fall through a distance h and the speed of the object as it falls through this distance are measured with a pendulum clock. The entire apparatus is taken on the moon and the experiment is repeated
(a) the measured times are same
(b) the measured speeds are same
(c) the actual times in the fall are equal
(d) the actual speeds are equal
A simple pendulum is constructed by hanging a heavy ball by a 5.0 m long string. It undergoes small oscillations. (a) How many oscillations does it make per second? (b) What will be the frequency if the system is taken on the moon where acceleration due to gravitation of the moon is 1.67 m/s2?
A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R. It makes small oscillations about the lowest point. Find the time period.
A simple pendulum of length 1 feet suspended from the ceiling of an elevator takes π/3 seconds to complete one oscillation. Find the acceleration of the elevator.
A closed circular wire hung on a nail in a wall undergoes small oscillations of amplitude 20 and time period 2 s. Find (a) the radius of the circular wire, (b) the speed of the particle farthest away from the point of suspension as it goes through its mean position, (c) the acceleration of this particle as it goes through its mean position and (d) the acceleration of this particle when it is at an extreme position. Take g = π2 m/s2.
A uniform disc of mass m and radius r is suspended through a wire attached to its centre. If the time period of the torsional oscillations be T, what is the torsional constant of the wire?
A simple pendulum of length l is suspended from the ceiling of a car moving with a speed v on a circular horizontal road of radius r. (a) Find the tension in the string when it is at rest with respect to the car. (b) Find the time period of small oscillation.
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 __________.
If the inertial mass and gravitational mass of the simple pendulum of length l are not equal, then the time period of the simple pendulum is
Write short notes on two springs connected in parallel.
The displacement of a particle is represented by the equation y = sin3ωt. The motion is ______.
The displacement of a particle varies with time according to the relation y = a sin ωt + b cos ωt.
Motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower point is ______.
- simple harmonic motion.
- non-periodic motion.
- periodic motion.
- periodic but not S.H.M.
The velocities of a particle in SHM at positions x1 and x2 are v1 and v2 respectively, its time period will be ______.
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 ______.
Which of the following expressions corresponds to simple harmonic motion along a straight line, where x is the displacement and a, b, and c are positive constants?
