Advertisements
Advertisements
प्रश्न
Using the differential equation of linear S.H.M., obtain an expression for acceleration, velocity, and displacement of simple harmonic motion.
Advertisements
उत्तर
- Expression for acceleration in linear S.H.M:
a. From the differential equation,
`("d"^2"x")/"dt"^2 + ω^2"x" = 0`
`("d"^2"x")/"dt"^2 = -ω^2"x"` ….(1)
b. But, linear acceleration is given by,
`("d"^2"x")/"dt"^2 = "a"` ...….(2)
From equations (1) and (2),
a = −ω2x .........….(3)
Equation (3) gives acceleration in linear S.H.M. - Expression for velocity in linear S.H.M:
a. From the differential equation of linear S.H.M
`("d"^2"x")/"dt"^2 = -ω^2"x"`
∴ `"d"/"dt"("dx"/"dt") = -ω^2"x"`
∴ `"dv"/"dt" = -ω^2"x"` ......`(∵ "dx"/"dt" = "v")`
∴ `"dv"/"dx"."dx"/"dt" = -ω^2"x"`
∴ `"v""dv"/"dx" = -ω^2"x"` ......`(∵ "dx"/"dt" = "v")`
∴ v dv = −ω2x dx ...........(4)
b. Integrating both sides of equation (4),
∫v dv = ∫-ω2x dx
`"v"^2/2 = -(ω^2"x"^2)/2` + C ............(5)
where, C is the constant of integration.
c. At extreme position, x = ± A and v = 0.
Substituting these values in equation (5),
0 = `-(ω^2"A"^2)/2 + "C"`
∴ C = `(ω^2"A"^2)/2` .............(6)
d. Substituting equation (6) in equation (5),
`"v"^2/2 = -(ω^2"x"^2)/2 + (ω^2"A"^2)/2`
∴ v2 = ω2A2 −ω2x2
∴ v2 = ω2(A2 - x2)
∴ v = ± ω`sqrt("A"^2 - "x"^2)`
This is the required expression for velocity in linear S.H.M. - Expression for displacement in linear S.H.M:
a. From the differential equation of linear S.H.M, velocity is given by,
v = ω`sqrt("A"^2 - "x"^2)` .....(1)
But, in linear motion, v = `"dx"/"dt"` ......(2)
From equation (1) and (2),
`"dx"/"dt" = ωsqrt("A"^2 - "x"^2)`
∴ `"dx"/sqrt("A"^2 - "x"^2)`= ω dt .....(3)
b. Integrating both sides of equation (3),
∫`"dx"/sqrt("A"^2 - "x"^2)` = ∫ω dt
∴ `sin^-1("x"/"A")` = ωt + Φ
where, α is constant of integration which depends upon initial condition (phase angle)
∴ `"x"/"A" = sin(ω"t" + Φ)`
∴ x = A sin(ωt + Φ)
This is the required expression for displacement of a particle performing linear S.H.M. at time t.
संबंधित प्रश्न
Choose the correct option:
The graph shows variation of displacement of a particle performing S.H.M. with time t. Which of the following statements is correct from the graph?
Find the change in length of a second’s pendulum, if the acceleration due to gravity at the place changes from 9.75 m/s2 to 9.8 m/s2.
A particle is performing simple harmonic motion with amplitude A and angular velocity ω. The ratio of maximum velocity to maximum acceleration is ______.
Acceleration of a particle executing S.H.M. at its mean position.
A particle is performing S.H.M. of amplitude 5 cm and period of 2s. Find the speed of the particle at a point where its acceleration is half of its maximum value.
For a particle performing SHM when displacement is x, the potential energy and restoring force acting on it is denoted by E and F, respectively. The relation between x, E and F is ____________.
Two identical wires of substances 'P' and 'Q ' are subjected to equal stretching force along the length. If the elongation of 'Q' is more than that of 'P', then ______.
In U.C.M., when time interval δt → 0, the angle between change in velocity (δv) and linear velocity (v) will be ______.
A particle is moving along a circular path of radius 6 m with a uniform speed of 8 m/s. The average acceleration when the particle completes one-half of the revolution is ______.
A body performing a simple harmonic motion has potential energy 'P1' at displacement 'x1' Its potential energy is 'P2' at displacement 'x2'. The potential energy 'P' at displacement (x1 + x2) is ________.
The relation between time and displacement for two particles is given by Y1 = 0.06 sin 27`pi` (0.04t + `phi_1`), y2 = 0.03sin 27`pi`(0.04t + `phi_2`). The ratio of the intensity of the waves produced by the vibrations of the two particles will be ______.
The phase difference between the instantaneous velocity and acceleration of a particle executing S.H.M is ____________.
If 'α' and 'β' are the maximum velocity and maximum acceleration respectively, of a particle performing linear simple harmonic motion, then the path length of the particle is _______.
Which of the following represents the acceleration versus displacement graph of SHM?
A particle executing S.H.M. has amplitude 0.01 m and frequency 60 Hz. The maximum acceleration of the particle is ____________.
A simple pendulum of length 'L' is suspended from a roof of a trolley. A trolley moves in horizontal direction with an acceleration 'a'. What would be the period of oscillation of a simple pendulum?
(g is acceleration due to gravity)
A block of mass 16 kg moving with velocity 4 m/s on a frictionless surface compresses an ideal spring and comes to rest. If force constant of the spring is 100 N/m then how much will be the spring compressed?
The displacement of a particle in S.H.M. is x = A cos `(omegat+pi/6).` Its speed will be maximum at time ______.
The displacements of two particles executing simple harmonic motion are represented as y1 = 2 sin (10t + θ) and y2 = 3 cos 10t. The phase difference between the velocities of these waves is ______.
A particle is performing SHM starting extreme position, graphical representation shows that between displacement and acceleration there is a phase difference of ______.
A particle of mass 5 kg moves in a circle of radius 20 cm. Its linear speed at a time t is given by v = 4t, t is in the second and v is in ms-1. Find the net force acting on the particle at t = 0.5 s.
A body of mass 0.5 kg travels in a straight line with velocity v = ax3/2 where a = 5 m–1/2s–1. The change in kinetic energy during its displacement from x = 0 to x = 2 m is ______.
Calculate the velocity of a particle performing S.H.M. after 1 second, if its displacement is given by x = `5sin((pit)/3)`m.
For a particle performing circular motion, when is its angular acceleration directed opposite to its angular velocity?
State the expressions for the displacement, velocity and acceleration draw performing linear SHM, starting from the positive extreme position. Hence, their graphs with respect to time.
A body is suspended from the two light springs separately, the periods of vertical oscillations are \[T_1\] and \[T_2\]. The same body is suspended from two springs connected in series first and then in parallel. The periods of oscillations in both cases respectively are ______.
A particle performs simple harmonic motion with amplitude A. Its speed is tripled at the instant that it is at a distance `(2 A)/3` from equilibrium position. The new amplitude of the motion is ______.
A pendulum is performing simple harmonic motion. The acceleration of the bob is 20 cm s−2 at a distance of 5 cm from mean position. The time period of oscillation is ______.
