A 103.0 g mass attached to a horizontal spring oscillates at a frequency of 2.52 Hz. At t = 0 s , the mass is at x = 6.31 cm and has a velocity of -11 cm/s.

Determine the phase constant in rad.

Determine the total energy in J.

Determine the position (in cm) at t = 1.55 s.

To determine the phase constant, total energy, and position at a given time for the oscillating mass-spring system, we can use various formulas and equations.

1. Phase constant (φ):
The equation for the position of an oscillating mass-spring system is given by:
x(t) = A * cos(ωt + φ)

where:
x(t) is the position of the mass at time t
A is the amplitude of the oscillation
ω is the angular frequency (ω = 2πf), where f is the frequency of oscillation
φ is the phase constant, which determines the initial position of the oscillating mass.

In this case, we are given the frequency f = 2.52 Hz. We can calculate the angular frequency ω as follows:
ω = 2πf = 2π * 2.52 Hz ≈ 15.85 rad/s

To determine the phase constant φ, we need to use the given information about the initial position and velocity of the mass:
At t = 0 s, the mass is at x = 6.31 cm and has a velocity of -11 cm/s.

First, we convert the velocity to the corresponding angular velocity (ω):
v = ω * A * sin(ωt + φ)
-11 cm/s = 15.85 rad/s * 6.31 cm * sin(φ)
solving for sin(φ), we get: sin(φ) ≈ -0.11

To determine the phase constant (φ), we use the inverse sine function (sin^(-1)):
φ ≈ sin^(-1)(-0.11)
Using a calculator, we find that φ ≈ -0.11 rad.

2. Total energy (E):
The total energy of an oscillating mass-spring system is the sum of its kinetic energy (KE) and potential energy (PE). It can be calculated using the following formula:

E = KE + PE = (1/2) * m * v^2 + (1/2) * k * x^2

where:
m is the mass (103.0 g = 0.103 kg)
v is the velocity of the mass (in m/s)
k is the spring constant (which depends on the specific spring used)
x is the displacement of the mass from its equilibrium position (in meters)

In this case, we are given the mass of the oscillating mass (m = 0.103 kg) and the velocity of the mass at t = 0 s (v = -11 cm/s). Note that the velocity needs to be converted to m/s:
v = -11 cm/s = -0.11 m/s

We also need the equilibrium position (x = 0) to calculate the total energy since only the displacement from the equilibrium position contributes to the potential energy.

Therefore, the total energy (E) at t = 0 s is:
E = (1/2) * m * v^2 + (1/2) * k * x^2
E = (1/2) * 0.103 kg * (-0.11 m/s)^2 + (1/2) * k * (0 m)^2
E = 0.0006 J + 0 J
E = 0.0006 J

3. Position at t = 1.55 s:
To determine the position of the mass at t = 1.55 s, we can use the equation for the position of an oscillating mass-spring system:
x(t) = A * cos(ωt + φ)

Given that we know the amplitude (A) and the phase constant (φ) from earlier calculations, we can plug in the values and solve for x(1.55 s):
x(1.55 s) = A * cos(ω * 1.55 s + φ)

However, the amplitude (A) is not given in the problem statement. Without knowing the amplitude, it is not possible to determine the position at a specific time accurately.

Hence, we can calculate the phase constant (φ ≈ -0.11 rad) and the total energy (E ≈ 0.0006 J), but we cannot determine the position at t = 1.55 s without knowing the amplitude (A) of the oscillation.