The position of an object connected to a spring varies with time according to the expression x = (5.0 cm) sin(9.4πt).
(a) Find the period of this motion.
(b) Find the frequency of the motion.
(c) Find the amplitude of the motion.
(d) Find the first time after t = 0 that the object reaches the position x = 2.6 cm.
Thank you!
(a) Find the period of this motion.
(b) Find the frequency of the motion.
(c) Find the amplitude of the motion.
(d) Find the first time after t = 0 that the object reaches the position x = 2.6 cm.
Thank you!
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Recognise that the expression is of the general form:
x = A.sin(ωt)
where A is the amplitude and ω is the angular frequency.
(c) By inspection, you can see that A = 5.0 cm
ω = 2πf
(b)
9.4πt = 2πft
9.4 = 2f
f = 9.4/2 = 4.7 Hz
(a) Period (T) = 1/f = 1/4.7 = 0.213 s
(d)
2.6 cm = (5.0 cm) sin(9.4πt).
2.6 / 5.0 = sin(9.4πt).
arc sin( 2.6 / 5.0) = 9.4πt
{arc sin may be more familiar to you as sin^-1. Remember we are working in RADIANS, not degrees!}
0.547 = 9.4πt
t = 0.547 /9.4π = 0.0186 s
rounding to 2 sig figures,
t = 0.019 s
x = A.sin(ωt)
where A is the amplitude and ω is the angular frequency.
(c) By inspection, you can see that A = 5.0 cm
ω = 2πf
(b)
9.4πt = 2πft
9.4 = 2f
f = 9.4/2 = 4.7 Hz
(a) Period (T) = 1/f = 1/4.7 = 0.213 s
(d)
2.6 cm = (5.0 cm) sin(9.4πt).
2.6 / 5.0 = sin(9.4πt).
arc sin( 2.6 / 5.0) = 9.4πt
{arc sin may be more familiar to you as sin^-1. Remember we are working in RADIANS, not degrees!}
0.547 = 9.4πt
t = 0.547 /9.4π = 0.0186 s
rounding to 2 sig figures,
t = 0.019 s
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x = (5.0 cm) sin(9.4πt)
It's a sin function, the max value of sin is 1 and the min is -1, therefore 1*5 and -1*5 will give you the higher and lowest values which are 5 and -5 in this case.
Using this, you can now graph a sin function and set the lower and higher values.
Next you want to equal the function to 0 (x=0), this will give you the values of t when distance to the spring is 0.
0=5 x sin (9.4 x pi x t)
This will throw infinite values but you only want the first 3 as the sin function only intercepts the x-axis on 3 locations per motion (360 degrees or 2 pi)
The values are 0 (of course), 6.09 (at the middle of the function) and 12.2 (at the end of the function).
With this values you now know that going through the movement takes about 12.2 seconds, this is the period of the motion.
The frequency of the motion comes from a formula:
f (frequency) = 1/T (period)
which is about 0.08 Hz
The amplitude from the motion (length) is from 5 to -5, which is 10 cm.
The first time after t = 0 that the object reaches the position x = 2.6 cm,
once more you equal the function to 2.6cm
2.6 = 5 sin(9.4 pi t)
and the result for t is 1.06 seconds.
So..
a.12.2 seconds
b. 0.08 Hz
c. 10 cm
d. 1.06 seconds
It's a sin function, the max value of sin is 1 and the min is -1, therefore 1*5 and -1*5 will give you the higher and lowest values which are 5 and -5 in this case.
Using this, you can now graph a sin function and set the lower and higher values.
Next you want to equal the function to 0 (x=0), this will give you the values of t when distance to the spring is 0.
0=5 x sin (9.4 x pi x t)
This will throw infinite values but you only want the first 3 as the sin function only intercepts the x-axis on 3 locations per motion (360 degrees or 2 pi)
The values are 0 (of course), 6.09 (at the middle of the function) and 12.2 (at the end of the function).
With this values you now know that going through the movement takes about 12.2 seconds, this is the period of the motion.
The frequency of the motion comes from a formula:
f (frequency) = 1/T (period)
which is about 0.08 Hz
The amplitude from the motion (length) is from 5 to -5, which is 10 cm.
The first time after t = 0 that the object reaches the position x = 2.6 cm,
once more you equal the function to 2.6cm
2.6 = 5 sin(9.4 pi t)
and the result for t is 1.06 seconds.
So..
a.12.2 seconds
b. 0.08 Hz
c. 10 cm
d. 1.06 seconds