The position of a mass oscillating on a spring is given by x = (7.0 cm) cos [2πt/(0.71 s)].
(a) What is the frequency of this motion?
(b) When is the mass first at the position x = -7.0 cm?
(a) Ah, the frequency of this motion can be found by using the formula: f = 1/T, where T is the period of the motion. In this case, we have T = 0.71 s, so plugging it into the formula gives us: f = 1/0.71 s. And voila, the frequency of this oscillation is approximately 1.41 Hz. It's swinging to its own tune!
(b) Well, to find when the mass is first at the position x = -7.0 cm, we need to find the time when the cosine function is equal to -1. Now, we know that the cosine function reaches its minimum value of -1 at an angle of 180 degrees or π radians. So, we'll set the argument of the cosine function equal to π and solve for t. Get ready for the math circus! Here we go:
(2πt)/(0.71 s) = π
Now, let's simplify this equation:
2πt = π(0.71 s)
We can cancel out those π's:
2t = 0.71 s
And solve for t:
t = 0.71 s / 2
t = 0.355 s
So, the mass is first at the position x = -7.0 cm at approximately 0.355 seconds into the motion. Ready or not, here it comes!
To find the frequency of the motion, we can use the equation:
frequency = 1 / period
(a) The period of oscillation can be found from the equation:
x = A * cos(2πt / T)
Comparing it with the given equation:
x = (7.0 cm) cos [2πt / (0.71 s)]
We can see that the period T is equal to 0.71 s.
Thus, the frequency is given by:
frequency = 1 / T = 1 / 0.71 s ≈ 1.41 Hz
So, the frequency of this motion is approximately 1.41 Hz.
(b) In order to find when the mass is at the position x = -7.0 cm, we can rearrange the equation as follows:
x = A * cos(2πt / T)
-7.0 cm = (7.0 cm) cos [2πt / (0.71 s)]
Let's solve for time (t):
cos [2πt / (0.71 s)] = -1
Using the inverse cosine function, we have:
2πt / (0.71 s) = π
Simplifying the equation:
t = (0.71 s) / 2 ≈ 0.355 s
So, the mass is first at the position x = -7.0 cm after approximately 0.355 seconds.
To find the answers to these questions, we'll need to use the given equation for the position of the mass on the spring:
x = (7.0 cm) cos [2πt/(0.71 s)]
(a) To find the frequency of the motion, we'll use the formula:
frequency (f) = 1 / period (T)
The period can be determined from the coefficient of t in the cosine function. In this case, it is 2π/0.71 s. So the period (T) is 0.71 s.
Now, we can calculate the frequency using the formula:
frequency (f) = 1 / period (T)
f = 1 / 0.71 s
f ≈ 1.41 Hz
Therefore, the frequency of this motion is approximately 1.41 Hz.
(b) To find when the mass first reaches the position x = -7.0 cm, we'll set the given equation equal to -7.0 cm and solve for t:
-7.0 cm = (7.0 cm) cos [2πt/(0.71 s)]
Now, we need to isolate the cosine term and solve for t. Divide both sides by 7.0 cm:
-1 = cos [2πt/(0.71 s)]
Next, take the inverse cosine (cos⁻¹) of both sides:
cos⁻¹(-1) = 2πt/(0.71 s)
The inverse cosine of -1 is π, so we have:
π = 2πt/(0.71 s)
Now, we can solve for t:
t = (0.71 s)(π / 2π)
t ≈ 0.71 s / 2
t ≈ 0.355 s
Therefore, the mass is first at the position x = -7.0 cm approximately 0.355 seconds after the start of the motion.