The position of an object connected to a spring varies with time according to the expression x = (7.1 cm) sin (2.1π t). Find
(a) the period of this motion (b) the frequency of the motion (c) the amplitude of the motion (d) the first time after t=0 that the object reaches the position x = 2.6 cm (Make sure your calculator is in radians mode)
(a)
Compare
x = (7.1 cm) sin (2.1π t)
with
x=Asin(ωt)
Then
ω= 2.1π,
=> T=2π/ ω =
= 2 π/2.1π=0.7 rad/s
(b)
f=1/T=1.42 Hz
(c)
A=7.1 cm
(d)
x=Asin(ωt)
2.6 = 7.1 sin (2.1π t)
sin (2.1π t) =2.6/7.1=0.366
(2.1π t)=sin⁻¹0.366=0.375 rad
t=0.375/2.1π=0.0569 s.
(a) The period of the motion can be found by using the formula T = 2π/ω, where T is the period and ω (omega) is the angular frequency. Looking at the expression x = (7.1 cm) sin (2.1π t), we can see that the coefficient of t inside the sine function is 2.1π. Comparing this with the general equation for angular frequency, ω = 2π/T, we can equate the two and solve for T:
2.1π = 2π/T
Simplifying, we find:
2.1/T = 1
Cross-multiplying, we get:
2.1 = T
Therefore, the period of this motion is 2.1 seconds.
(b) The frequency can be found using the formula f = 1/T, where f is the frequency and T is the period. By substituting the value we found for the period, we can calculate the frequency:
f = 1/2.1
f ≈ 0.476 Hz
Therefore, the frequency of this motion is approximately 0.476 Hz.
(c) The amplitude of the motion can be determined by looking at the coefficient in front of the sine function. In this case, the amplitude is 7.1 cm.
Therefore, the amplitude of this motion is 7.1 cm.
(d) To find the first time after t=0 that the object reaches the position x = 2.6 cm, we can set up the equation:
2.6 = 7.1 sin (2.1π t)
Dividing both sides by 7.1, we have:
sin (2.1π t) = 0.3661971831
To find the time t that satisfies this equation, we can use the arcsine function:
2.1π t = arcsin(0.3661971831)
Dividing both sides by 2.1π, we find:
t = arcsin(0.3661971831)/(2.1π)
Using a calculator in radian mode, we can solve for t and find the first time after t=0:
t ≈ 0.09308057 seconds
To solve this problem, let's go step-by-step:
(a) To find the period of the motion, we use the formula T = 2π/ω, where T is the period and ω is the angular frequency.
In this case, the expression for the position of the object is x = (7.1 cm) sin (2.1π t), where t represents time.
Comparing this expression to the standard form of the equation for simple harmonic motion, x = A sin (ωt), we can see that ω = 2.1π.
Using the formula T = 2π/ω, we can calculate the period:
T = 2π / (2.1π) = 1 / 2.1 ≈ 0.476 seconds.
Therefore, the period of this motion is approximately 0.476 seconds.
(b) The frequency of the motion is the reciprocal of the period. So, the frequency is given by f = 1/T.
Substituting the value of T, we get:
f = 1 / 0.476 ≈ 2.101 Hz.
Therefore, the frequency of the motion is approximately 2.101 Hz.
(c) The amplitude of the motion is given by the coefficient A in the equation x = A sin (ωt).
From the given expression, we can see that the amplitude is A = 7.1 cm.
Therefore, the amplitude of the motion is 7.1 cm.
(d) To find the first time after t = 0 that the object reaches the position x = 2.6 cm, we need to solve the equation for t.
The equation for the position of the object is x = (7.1 cm) sin (2.1π t).
Setting x equal to 2.6 cm, we have:
2.6 = 7.1 sin (2.1π t).
Now, we need to rearrange the equation to solve for t. Divide both sides by 7.1:
sin (2.1π t) = 2.6 / 7.1.
Next, take the inverse sine (sin⁻¹) of both sides to isolate t:
2.1π t = sin⁻¹ (2.6 / 7.1).
Finally, divide both sides by 2.1π to solve for t:
t = sin⁻¹ (2.6 / 7.1) / (2.1π).
Using a calculator in radians mode, you can find the value of t.
To determine the period of the motion, we can look at the equation: x = (7.1 cm) sin (2.1π t). The argument of the sine function inside the parentheses is 2.1π t.
(a) The period of the motion is the time it takes for the object to complete one full oscillation. It is given by the formula T = 2π/ω, where ω is the angular frequency.
In this case, the angular frequency is the coefficient in front of t, which is 2.1π. So, we can substitute this value into the formula:
T = 2π / (2.1π) = 2 / 2.1 ≈ 0.9524 s
Therefore, the period of the motion is approximately 0.9524 seconds.
(b) The frequency of the motion is the reciprocal of the period. So, we can calculate it as f = 1 / T:
f = 1 / 0.9524 ≈ 1.05 Hz
Therefore, the frequency of the motion is approximately 1.05 Hz.
(c) The amplitude of the motion is the maximum displacement from the equilibrium position. In this case, the amplitude is given by the coefficient in front of the sine function, which is 7.1 cm.
Therefore, the amplitude of the motion is 7.1 cm.
(d) To find the first time after t = 0 that the object reaches the position x = 2.6 cm, we can set the equation x = 2.6 cm and solve for t:
2.6 = 7.1 sin (2.1π t)
Dividing both sides by 7.1, we have:
sin (2.1π t) = 2.6 / 7.1
To find the value of t, we need to take the inverse sine of both sides. However, make sure your calculator is in radians mode.
π t = arcsin (2.6 / 7.1)
Now, we can solve for t by dividing both sides by π:
t = arcsin (2.6 / 7.1) / π
Use a calculator to find the approximate value of arcsin (2.6 / 7.1), and then divide it by π to get the value of t.
Note: The unit for time will be the same as the unit used in the period calculation, which is seconds.
By following these steps, you can find the answers to all the questions.