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.
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 ______.
A wheel of M.I. 50 kg m2 starts rotating on applying a constant torque of 200 Nm. Its angular velocity after 2.5 second from the start 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 ____________.
A particle executing S.H.M. has amplitude 0.01 m and frequency 60 Hz. The maximum acceleration of the particle is ____________.
A body of mass 5 g is in S.H.M. about a point with amplitude 10 cm. Its maximum velocity is 100 cm/s. Its velocity will be 50 cm/s at a distance of, ____________.
The displacement of a particle is 'y' = 2 sin `[(pit)/2 + phi]`, where 'y' is cm and 't' in second. What is the maximum acceleration of the particle executing simple harmonic motion?
(Φ = phase difference)
The maximum speed of a particle in S.H.M. is 'V'. The average speed is ______
A body is executing S.H.M. Its potential energy is E1 and E2 at displacements x and y respectively. The potential energy at displacement (x + y) 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)
The bob of a simple pendulum is released at time t = 0 from a position of small angular displacement. Its linear displacement is ______.
(l = length of simple pendulum and g = acceleration due to gravity, A = amplitude of S.H.M.)
A body perform linear simple harmonic motion of amplitude 'A'. At what displacement from the mean position, the potential energy of the body is one fourth of its total energy?
The displacement of a particle is represented by the equation `y = 3 cos (pi/4 - 2ωt)`. The motion of the particle 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 performs linear SHM at a particular instant, velocity of the particle is 'u' and acceleration is a while at another instant velocity is 'v' and acceleration is 'β (0 < α < β). The distance between the two position is ______.
In figure, a particle is placed at the highest point A of a smooth sphere of radius r. It is given slight push and it leaves the sphere at B, at a depth h vertically below A, such that h is equal to ______.
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.
The displacement of a particle of mass 3 g executing simple harmonic motion is given by Y = 3 sin (0.2 t) in SI units. The kinetic energy of the particle at a point which is at a distance equal to `1/3` of its amplitude from its mean position 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.
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 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 ______.
