To find the phase constant $\phi$ for the harmonic oscillator given the velocity function $v(t)$ and position function $x(t) = x_m \cos(\omega t + \phi)$, we need to use the relationship between position and velocity in harmonic motion.

Given Information:

• Position function: $x(t) = x_m \cos(\omega t + \phi)$.
• The vertical axis scale for $v(t)$ is set by $v_s = 3.20 \, \text{cm/s}$.

Solution Outline:

1. Find the relationship between $v(t)$ and $x(t)$:
   For simple harmonic motion, the velocity is the time derivative of the position: $v(t) = \frac{dx}{dt} = -x_m \omega \sin(\omega t + \phi)$

2. Determine $v(t)$ characteristics from the graph:
   Observing the graph (though details aren't fully clear), we should note the points where $v(t)$ is zero and the phase of $v(t)$ with respect to $x(t)$. If $v(t)$ has a peak at a certain point, we can infer the phase constant $\phi$ based on the typical behavior of sine and cosine functions.

3. Analyze the phase shift $\phi$:
   The sign and amplitude of $v(t)$ will help determine the phase constant. If, for example, $v(t)$ reaches a maximum or minimum when $x(t) = 0$, $\phi$ can be found based on the expected phase relationships of sine and cosine.

Since more specific points on the graph are not visible, could you clarify or provide details about where $v(t)$ crosses zero or peaks relative to $x(t)$? With these details, I can accurately calculate the phase constant $\phi$.

Would you like more information on phase shifts or on the properties of harmonic oscillations?

Related Questions:

1. What are typical phase relationships between velocity and position in harmonic motion?
2. How does the amplitude of velocity relate to the amplitude of position in a harmonic oscillator?
3. What other physical quantities can be determined from knowing the phase constant?
4. How does angular frequency $\omega$ influence the phase shift in simple harmonic motion?
5. How would you determine $\phi$ if the position and velocity functions had different forms?

Tip: Remember, in harmonic motion, when $x(t)$ is at a maximum, $v(t)$ is zero, and when $v(t)$ is at a maximum or minimum, $x(t)$ passes through zero.