A=A'sin(wt) using this equation

time taken=T/30, where T is the time period of SHM
USE W= 2π/T

Understanding SHM
In Simple Harmonic Motion (SHM), a particle oscillates between two extremes defined by its amplitude (A) and has a specific time period (T). The particle's position as a function of time can be described by a sinusoidal function.
Initial Conditions
- The particle starts from rest at the maximum displacement (x = A).
- The motion begins when the particle is at its extreme position.
Distance Calculation
To find the time taken to travel a distance of A/5, we need to understand the particle's motion during the first part of its journey.
- The total distance to be covered is A/5, starting from x = A.
- The motion can be analyzed using the equations of SHM.
Velocity in SHM
The velocity (v) of a particle in SHM is given by the formula:
- v = ω√(A² - x²)
where ω = 2π/T is the angular frequency.
Time Calculation
1. Initial position: x = A
2. Final position: x = A - A/5 = (4/5)A
3. Calculate velocity at x = (4/5)A:
- v = ω√(A² - (4/5)²A²)
- v = ω√(1 - (16/25)) = ω√(9/25) = (3/5)ω
4. Average velocity: Since the particle starts from rest, average velocity can be approximated as:
- v_avg = (initial v + final v) / 2 = (0 + (3/5)ω) / 2 = (3/10)ω
5. Time taken (t):
- Distance = v_avg × t
- A/5 = (3/10)ω × t
- t = (A/5) / (3/10)ω = (2A/15) × (T/2π)
6. Final result:
- t = (2AT)/(15π)
Using these steps, you can determine the time required for the particle to travel a distance of A/5 while performing SHM.