The position of a mass oscillating on a spring is given by x=(4.7 cm)cos [2pi t/(0.52 s)].
What is the period of this motion?
What is the first time the mass is at the position x=0?
The period of this motion is 0.52 seconds. The first time the mass is at the position x=0 is at t=0.
To find the period of the motion, we can use the formula:
T = 2π / ω,
where T is the period and ω is the angular frequency.
In the given equation, we can see that the angular frequency is given by:
ω = 2π / T,
Since we're given ω = 2π / (0.52 s), we can solve for T:
T = 2π / ω = 2π / (2π / (0.52 s)) = 0.52 s.
Therefore, the period of this motion is 0.52 s.
To find the first time the mass is at the position x = 0, we can set the equation equal to zero and solve for t:
0 = (4.7 cm)cos [2πt / (0.52 s)].
Using the cosine function, we know that when the argument inside the cos function is 0, the cosine value is equal to 1. Thus, we need to find when the argument is equal to 0:
2πt / (0.52 s) = 0.
To solve for t, we rearrange the equation:
t = 0.
Therefore, the first time the mass is at the position x = 0 is when t = 0.
To find the period of the motion, we need to know the formula for the period of an oscillating mass on a spring.
The period (T) is given by the formula:
T = (2π) / ω
Where ω is the angular frequency, defined as ω = 2πf, and f is the frequency.
In the given equation, x = (4.7 cm)cos [2πt / (0.52 s)], we can see that the angular frequency (ω) is 2π / (0.52 s).
Substituting this value into the formula for the period:
T = (2π) / (2π / 0.52 s)
T = 0.52 s
Therefore, the period of this motion is 0.52 seconds.
To find the first time the mass is at the position x = 0, we can set the equation equal to zero and solve for time (t).
Setting x = 0, we have:
0 = (4.7 cm)cos [2πt / (0.52 s)]
Taking the inverse cosine of both sides, we get:
cos [2πt / (0.52 s)] = 0
Since cosθ = 0 when θ = (2n + 1)π/2, where n is an integer, we can solve for t as:
2πt / (0.52 s) = (2n + 1)π/2
Simplifying:
t = (0.52 s / 2π) * [(2n + 1)π/2]
Now we can solve for the first value of t when x = 0 by plugging in n = 0:
t = (0.52 s / 2π) * [(2(0) + 1)π/2]
t = (0.52 s / 2π) * (π/2)
t = 0.13 s
Therefore, the first time the mass is at the position x = 0 is 0.13 seconds.