A particle is moving along the x-axis under the influence of a force given by F=-5x+15,at t=0, the particle is located at x=6, ant is having zero velocity it takes 0.5 second to reach the origin for the first time the equation of motion of the particle can be represented by
Answers
The equation of motion is 3 +3 cos (2πt)
its mean position is 3 as the ant is located at x=6
therefore its amplitude is =3
given- to reach origin it takes 0.5 sec so
T=1sec
x=3 + 3sin ( wt + ¢)
taking t=0
3+3 sin ¢= 6
3sin¢=6-3
sin¢=1
¢=π/2
x=3+3sin(wt+π/2)
=3+ 3coswt
=3 + 3cos (2πt)
The equation of motion of the particle can be represented by 3 + 3 cos(2πt)
Explanation:
From question, the equation of mean position of the particle is given as:
F = -5x + 15 = 0
Thus, the mean position of the particle is at ⇒ x 3
From question, the extreme position of the particle is at ⇒ x = 6
Hence, it is a simple harmonic motion. This motion takes place at x = 0.
The time taken by the particle to move from x = 6 to x = 0 is 0.5 seconds.
Now, the time period is:
T/2 = 0.5 seconds
∴ T = 1 second
Now, the equation of the motion is given as:
x = 3 + A(sin ωt + Φ₀)
Where,
A = Amplitude = 3
ω = 2π × 1 = 2π
Φ₀ = π/2 × 5ω
Now, the equation of motion is:
∴ x = 3 + 3 cos (2πt)