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.

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