For a particle vibrating simple harmonically, the displacement is 15cm at the instant the velocity is 5cm/sec and the displacement is 8cm at the instant the velocity is 18cm/sec.
Calculate i)Amplitude
ii) Frequency
iii) The time period
Good
To calculate the amplitude, we can use the formula:
Amplitude = maximum displacement
Given that the displacement is 15cm, the amplitude is 15cm.
To calculate the frequency, we can use the formula:
Frequency = (velocity / wavelength)
However, we don't have the wavelength directly. We can use the fact that the velocity is related to the amplitude and frequency by the equation:
velocity = 2 * π * frequency * amplitude
Given that the velocity is 5cm/sec, we can substitute the values and solve for frequency:
5cm/sec = 2 * π * frequency * 15cm
Rearranging the equation, we get:
frequency = 5cm/sec / (2 * π * 15cm) ≈ 0.053 Hz
Therefore, the frequency is approximately 0.053 Hz.
To calculate the time period, we can use the formula:
Time period = 1 / Frequency
Given that the frequency is 0.053 Hz, we can substitute the value and solve for the time period:
Time period = 1 / 0.053 Hz ≈ 18.868 seconds
Therefore, the time period is approximately 18.868 seconds.
To find the amplitude, we can utilize the relationship between displacement and velocity for an object undergoing simple harmonic motion. The amplitude represents the maximum displacement from the equilibrium position. We can identify the amplitude using the given information about displacement and velocity.
To determine the amplitude:
1) Recall that at any point during simple harmonic motion, the displacement and velocity are related by the equation: v = ω√(A² - x²), where v represents velocity, A represents the amplitude, x represents displacement, and ω is the angular frequency.
2) We are given two pairs of values: (x₁, v₁) = (15 cm, 5 cm/sec) and (x₂, v₂) = (8 cm, 18 cm/sec). We can utilize these values to create two equations:
For the first pair: 5 = ω√(A² - 15²)
For the second pair: 18 = ω√(A² - 8²)
3) Solve these two equations simultaneously to find the value of A, which represents the amplitude.
To determine the frequency:
1) The frequency (f) represents the number of oscillations per unit time. It is related to the angular frequency (ω) by the equation: ω = 2πf.
2) Once we have the value of ω, we can use it to find the frequency.
To determine the time period:
1) The time period (T) represents the duration of one complete oscillation. It is the inverse of the frequency: T = 1/f.
Using these steps, let's calculate the amplitude, frequency, and time period:
i) To find the amplitude:
- Square the equations obtained from step 2 above to eliminate the square root:
For the first pair: (5/ω)² = A² - 15²
For the second pair: (18/ω)² = A² - 8²
- Apply substitution:
From the first equation: (5/ω)² = A² - 225
From the second equation: (18/ω)² = A² - 64
- Equate the two equations:
A² - 225 = A² - 64
- Simplify and solve for A:
225 - 64 = 0
A² = 225 - 64
A² = 161
A = √161 cm
ii) To find the frequency:
- Recall that ω = 2πf, so we need to find the value of ω first.
- Use either of the equations above (we'll use the first one):
(5/ω)² = (√161)² - 15²
25/ω² = 161 - 225
25/ω² = -64
ω² = -25/64
ω = √(-25/64) cm/sec
- Convert ω to frequency using the equation ω = 2πf:
√(-25/64) = 2πf
- Solve for f:
f = √(-25/64) / (2π)
iii) To find the time period:
- Recall that the time period (T) is the inverse of the frequency: T = 1/f.
- Use the value of f obtained in the previous step and calculate T.