Advertisements
Advertisements
प्रश्न
Two parallel S.H.M.s represented by `"x"_1 = 5 sin(4π"t" + π//3)` cm and `"x"_2 = 3sin (4π"t" + π//4)` cm are superposed on a particle. Determine the amplitude and epoch of the resultant S.H.M.
Advertisements
उत्तर
Given:
Two parallel S.H.Ms represented by `"x"_1 = 5 sin(4π"t" + π//3)` cm and `"x"_2 = 3 sin(4π"t" + π//4)` cm are superposed on a particle.
To find: The amplitude and epoch of the resultant SHM.
using superposition principle,
amplitude of resultant is given by,
A = `sqrt("A"_1^2 + "A"_2^2 + 2"A"_1"A"_2 "cos" phi)`
here, A₁ = 5, A₂ = 3 and Φ = `π/3 - π/4 = π/12`
now A = `sqrt(5^2 + 3^2 + 2(5)(3)cos(π/12))`
`= sqrt(25 + 9 + 30 xx 0.966)`
`= sqrt(62.98)`
= 7.936 cm
epoch of the resultant, θ = tan¯¹ `[("A"_1 "sin" Φ_1 + "A"_2 "sin" Φ_2)/("A"_1 "cos" Φ_1 + "A"_2 cos Φ_2)]`
= tan¯¹ `[(5sin(π/3) + 3sin(π/4))/(5cos(π/3) + 3cos(π/4))]`
= tan¯¹`[(5 × sqrt3/2 + 3 × 1/sqrt2)/(5 × 1/2 + 3 × 1/sqrt2)]`
= tan¯¹ `[(5 sqrt3 + 3 sqrt2)/(5 + 3sqrt2)]`
= 54° 23'
Therefore, the amplitude is 7.936 cm, and the epoch of the resultant is 54°23'.
APPEARS IN
संबंधित प्रश्न
Show that a linear S.H.M. is the projection of a U.C.M. along any of its diameter.
Define linear simple harmonic motion.
Choose the correct option:
A body of mass 1 kg is performing linear S.H.M. Its displacement x (cm) at t(second) is given by x = 6 sin `(100t + π/4)`. Maximum kinetic energy of the body is ______.
What does the phase of π/2 indicate in linear S.H.M.?
Define linear S.H.M.
The equation of a simple harmonic motion is given by, x = 8 sin (8πt) + 6 cos (8πt), the initial phase angle is ______
At extreme positions of a particle executing simple harmonic motion, ______
In a spring-block system, length of the spring is increased by 5%. The time period will ____________.
A particle executes simple harmonic motion and is located at x = a, b, and c at times t0, 2t0, and 3t0 respectively. The frequency of the oscillation is ______.
For a particle executing simple harmonic motion, which of the following statements is NOT correct?
A simple pendulum performs simple harmonic motion about x = 0 with an amplitude A and time period T. The speed of the pendulum at x =A/2 will be ____________.
The velocities of a particle performing linear S.H.M are 0.13 m/s and 0.12 m/s, when it is at 0.12 m and 0.13 m from the mean position respectively. If the body starts from mean position, the equation of motion is ____________.
A particle executes simple harmonic motion with amplitude 'A' and period 'T'. If it is halfway between mean position and extreme position, then its speed at that point is ______.
The equation of a particle executing simple harmonic motion is given by x = sin π `("t" + 1/3)` m. At t = 1s, the speed of particle will be ______. (Given π = 3.14)
Two simple harmonic motions are represented by the equations y1 = 0.1 sin `(100pi"t"+pi/3)` and y1 = 0.1 cos πt.
The phase difference of the velocity of particle 1 with respect to the velocity of particle 2 is ______.
A particle is executing simple harmonic motion with amplitude A. When the ratio of its kinetic energy to the potential energy is `1/4`, its displacement from its mean position is ______.
A light rod of length 2m suspended from the ceiling horizontally by means of two vertical wires of equal length. A weight W is hung from a light rod as shown in figure.
The rod hung by means of a steel wire of cross-sectional area A1 = 0.1 cm2 and brass wire of cross-sectional area A2 = 0.2 cm2. To have equal stress in both wires, T1/T2 = ______.
For a particle performing linear S.H.M., its average speed over one oscillation is ______. (a = amplitude of S.H.M., n = frequency of oscillation)
If a body is executing simple harmonic motion, then ______.
Two simple harmonic motion are represented by the equations, y1 = 10 sin `(3pi"t"+pi/4)` and y2 = 5`(3sin3pi"t"+sqrt3cos3pi"t")`. Their amplitudes are in the ratio of ______.
What do you know about restoring force?
The displacement of a particle performing simple harmonic motion is `1/3` rd of its amplitude. What fraction of total energy will be its kinetic energy?
