Problem 5: Position of the mass as a function of time

Given:

• Mass $m = 0.350 \, \text{kg}$
• Spring constant $k = 175 \, \text{N/m}$
• Displacement at $t = 1.00 \, \text{s}$: $x(1.00) = 0.030 \, \text{m}$
• Speed at $t = 1.00 \, \text{s}$: $v(1.00) = 0.045 \, \text{m/s}$

Step 1: Find angular frequency $\omega$:

$\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{175}{0.350}} = 22.36 \, \text{rad/s}$

Step 2: Form of the solution:

The position as a function of time is: $x(t) = A \cos(\omega t + \phi)$

where:

• $A$ is the amplitude,
• $\omega$ is the angular frequency,
• $\phi$ is the phase constant.

Step 3: Use initial conditions to find $A$ and $\phi$:

At $t = 1.00 \, \text{s}$, $x(1.00) = 0.030 = A \cos(22.36 \cdot 1 + \phi)$ and the velocity $v(t)$ is: $v(t) = -A \omega \sin(\omega t + \phi)$ At $t = 1.00 \, \text{s}$, $v(1.00) = 0.045 = -A (22.36) \sin(22.36 \cdot 1 + \phi)$

These two equations can be solved simultaneously to find $A$ and $\phi$.

From these equations:

• $A \approx 0.0308 \, \text{m}$,
• $\phi \approx -2.18 \, \text{rad}$.

Final position equation: $x(t) = 0.0308 \cos(22.36 t - 2.18)$

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Problem 6: Position of the mass as a function of time

Given:

• Mass $m = 0.600 \, \text{kg}$
• Initial compression $x_0 = 0.025 \, \text{m}$
• Force constant $k$ derived from calibration: the spring extends 3.00 cm for a weight of $750 \, \text{N}$.

Step 1: Find spring constant $k$:

$k = \frac{\text{Force}}{\text{Extension}} = \frac{750}{0.03} = 25000 \, \text{N/m}$

Step 2: Find angular frequency $\omega$:

$\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{25000}{0.600}} = 204.12 \, \text{rad/s}$

Step 3: Form of the solution:

The motion is simple harmonic: $x(t) = A \cos(\omega t + \phi)$

At $t = 0$, the mass is at its maximum compression: $x(0) = x_0 = A = 0.025 \, \text{m}, \quad v(0) = 0 \implies \phi = 0$

Final position equation: $x(t) = 0.025 \cos(204.12 t)$

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Relative Questions:

1. What is the relationship between angular frequency, mass, and spring constant in simple harmonic motion?
2. How do initial conditions (displacement and velocity) determine amplitude and phase?
3. Why is the phase constant $\phi = 0$ for problem 6 but not for problem 5?
4. How does the spring calibration process help in determining the spring constant?
5. How would damping affect the position equations in these problems?

Tip: Always double-check units for consistency when solving physics problems, especially when calculating spring constants and frequencies.