A mass is oscillating on a spring with a period of 4.60s. At t=0s the mass has zero speed is at x=8.30cm. What is the value of t the first time after t=0s that the mass is at x=4.15 cm?
To solve this problem, we can use the equation for the displacement of a mass-spring system:
x(t) = A * cos(ωt + φ)
Where:
x(t) is the displacement of the mass at time t
A is the amplitude of the oscillation
ω is the angular frequency of the oscillation
φ is the phase angle
Given that the period of the oscillation is 4.60s, we can calculate the angular frequency:
T = 2π / ω
4.60s = 2π / ω
ω = 2π / 4.60s
Now, let's find the phase angle φ using the initial conditions:
At t = 0s, x = 8.30cm
8.30cm = A * cos(0 + φ)
cos(φ) = 8.30cm / A ---(1)
Now, to find the value of A, we can use the fact that the maximum displacement is equal to the amplitude:
A = 8.30cm
Substituting this back into equation (1):
cos(φ) = 8.30cm / 8.30cm
cos(φ) = 1
φ = 0 radians
With the phase angle φ, we can now find the time t at which the mass is at x = 4.15cm:
x(t) = 8.30cm * cos(ωt + 0)
4.15cm = 8.30cm * cos(ωt)
cos(ωt) = 4.15cm / 8.30cm
cos(ωt) = 0.5
To find the angle whose cosine is 0.5, we can use the inverse cosine function (arccos):
ωt = arccos(0.5)
ωt = π / 3
Finally, we can solve for t:
ωt = π / 3
t = (π / 3) / ω
Substituting the value of ω we found earlier:
t = (π / 3) / (2π / 4.60s)
t ≈ 0.384s
Therefore, the first time after t = 0s that the mass is at x = 4.15cm is approximately 0.384s.
To find the value of t when the mass is at x = 4.15 cm, we need to use the equation for the displacement of an object undergoing simple harmonic motion:
x(t) = A * cos(ωt + φ)
Where:
- x(t) is the displacement of the mass at time t
- A is the amplitude of the oscillation (maximum displacement)
- ω is the angular frequency of the oscillation
- φ is the phase constant
In this case, we are provided with the amplitude (A = 8.30 cm), and we need to find the time (t) when the mass is at x = 4.15 cm.
First, let's find the angular frequency (ω) using the formula:
ω = 2π / T
Where:
- T is the period of the oscillation
Given that the period (T) is 4.60 s, we can calculate ω:
ω = 2π / 4.60 s
Next, we can rearrange the displacement equation to solve for time (t):
cos(ωt + φ) = x / A
Substituting the provided values (x = 4.15 cm and A = 8.30 cm), we have:
cos(ωt + φ) = 4.15 cm / 8.30 cm
Now, let's solve for ωt + φ using the inverse cosine function (cos⁻¹) on both sides of the equation:
ωt + φ = cos⁻¹(4.15 cm / 8.30 cm)
Finally, we can solve for t by rearranging the equation:
t = (cos⁻¹(4.15 cm / 8.30 cm) - φ) / ω
As we don't have the phase constant (φ) provided in the question, we can assume φ = 0 since at t = 0s, the mass has zero speed.
Using the calculated values of ω and the provided values, we can substitute them into the equation to find t:
t = (cos⁻¹(4.15 cm / 8.30 cm) - 0) / ω
Evaluate this expression to find the value of t.