At time an object of mass 50.34 g on the end of a horizontal spring is moving to the right at 10 cm/s and is displaced 2 cm to the left from the equilibrium position. If the spring constant is 0.234N/m, find an expression for the position of the particle as a function of time.

To find the expression for the position of the particle as a function of time, we can use the equation for simple harmonic motion (SHM).

The equation for SHM is given by:

x(t) = A * cos(wt + phi)

Where:
x(t) is the position of the particle at time t
A is the amplitude of the oscillation
w is the angular frequency of the oscillation
t is the time
phi is the phase constant

In this case, the amplitude is given by the displacement from the equilibrium position, which is 2 cm (or 0.02 m). The angular frequency is given by:

w = sqrt(k / m)

Where:
k is the spring constant (0.234 N/m)
m is the mass of the object (50.34 g or 0.05034 kg)

Let's calculate the angular frequency:

w = sqrt(0.234 N/m / 0.05034 kg)
w ≈ 9.661 rad/s

Now we need to determine the phase constant phi. This can be determined by considering the initial conditions of the problem. The object is moving to the right at 10 cm/s, which means it has an initial velocity of +10 cm/s (or 0.1 m/s). Since the object is initially displaced 2 cm to the left, the phase constant phi can be determined using the displacement and velocity:

x(0) = A * cos(phi) = -0.02 m (2 cm to the left)
v(0) = -A * w * sin(phi) = 0.1 m/s (velocity to the right)

From these equations, we can solve for phi:

cos(phi) = -0.02 / A
sin(phi) = 0.1 / (-A * w)

Taking the ratio of these two equations, we get:

tan(phi) = (0.1 / (-A * w)) / (-0.02 / A)
tan(phi) = 5 / w

Now we can calculate the phase constant phi:

phi = atan(5 / w)
phi ≈ 0.497 radians

Finally, we can write the expression for the position of the particle as a function of time:

x(t) = A * cos(wt + phi)
x(t) = 0.02 * cos(9.661t + 0.497)

Thus, the expression for the position of the particle as a function of time is x(t) = 0.02 * cos(9.661t + 0.497).