An object suspended from a spring is oscillating up and down. The distance from the high point to the low point is 30 cm, and the objects take 4 sec to complete 5 cycles.For the first few cycles the distance from the mean position, d(t), in cm with respect to the time t sec is modeled by a sin function.
a)write an equation the describes the distance of the object from its mean position as a function if time.
general shape
d = a sin kt
we know a = 15
and if it takes 4 sec to complete 5 cycles
then 1 cycle or 1 period = 4/5 seconds
to find k,
2π/k= 4/5
4k = 10π
k = 5π/2
equation:
d = 15 sin (5π/2) t
Well, if the distance from the high point to the low point is 30 cm, then the amplitude of the oscillation is half of that, which is 15 cm.
If we assume that the mean position is at 0 cm, then the equation that describes the distance of the object from its mean position as a function of time can be written as:
d(t) = 15*sin(ωt)
Where ω is the angular frequency, given by ω = 2πf, with f being the frequency.
Since the object completes 5 cycles in 4 seconds, we can calculate the frequency as:
f = 5 cycles / 4 sec
Substituting this value into the angular frequency equation, we get:
ω = 2π (5/4) = 5π/2
Therefore, the equation that describes the distance of the object from its mean position as a function of time is:
d(t) = 15*sin((5π/2)t)
The equation that describes the distance of the object from its mean position as a function of time can be written as:
d(t) = A * sin(ωt + φ) + C
where:
- d(t) is the distance from the mean position at time t (in cm),
- A is the amplitude of the oscillation (half the distance from the high point to the low point, in cm),
- ω is the angular frequency of the oscillation (in radians per second),
- φ is the phase shift (in radians), and
- C is the vertical shift (mean position, in cm).
In this case, we are given that the distance from the high point to the low point is 30 cm (so the amplitude is 15 cm), and the object takes 4 seconds to complete 5 cycles.
To find the values of ω and φ, we can use the formula:
ω = 2π / T
where T is the period (in seconds) and is given by:
T = time taken / number of cycles
Here, the time taken is 4 seconds and the number of cycles is 5. Substituting these values into the formula, we get:
T = 4 / 5 = 0.8 seconds
Therefore, the value of ω is:
ω = 2π / 0.8 = 7.85 radians per second
As for the phase shift φ, we are not given any information in the question. To find its value, we would need additional information.
So, the equation that describes the distance of the object from its mean position as a function of time is:
d(t) = 15 * sin(7.85t + φ) + C
To write an equation that describes the distance of the object from its mean position as a function of time, we can use a sine function, as mentioned in the question.
First, let's understand the properties of the sine function and how they relate to the given information:
1. Amplitude (A): The amplitude of a sine function represents half the distance between the highest point and the lowest point. In this case, the distance from the high point to the low point is given as 30 cm, so the amplitude would be half of that, which is 15 cm.
2. Period (T): The period of a sine function represents the time taken to complete one full cycle. It is the reciprocal of the frequency (f). We are given that the object takes 4 seconds to complete 5 cycles. Therefore, the time period for one cycle (T) can be calculated by dividing the total time taken (4 seconds) by the number of cycles (5). Thus, T = (4 seconds) / (5 cycles) = 0.8 seconds per cycle.
3. Phase Shift (ϕ): The phase shift represents any horizontal shift of the sine function. In this case, we are not given any information about a phase shift, so we can assume it to be zero.
The equation that describes the distance of the object from its mean position as a function of time (t) can be written as:
d(t) = A * sin(2π/T * (t - ϕ))
Plugging in the values we obtained:
d(t) = 15 * sin(2π/0.8 * t)
Simplifying:
d(t) = 15 * sin(2.5π * t)
Therefore, the equation that describes the distance of the object from its mean position as a function of time is d(t) = 15 * sin(2.5π * t).