Characteristic quantities in simple harmonic motion

To solve these simple harmonic motion problems, we'll use the following concepts:

### Problem 1:

**a) Amplitude**:
The amplitude is the maximum displacement from the equilibrium position.
Given: Maximum displacement = 0.05 m
So, the amplitude \( A = 0.05 \, \text{m} \).

**b) Period**:
The period \( T \) is given by the formula:
\[ T = \frac{2\pi}{\omega} \]
where \( \omega \) is the angular frequency.
Given: \( \omega = 10 \, \text{rad/s} \)
\[ T = \frac{2\pi}{10} = \frac{\pi}{5} \, \text{s} \]

**c) Maximum Velocity**:
The maximum velocity \( v_{\text{max}} \) can be calculated using:
\[ v_{\text{max}} = A \omega \]
Given: \( A = 0.05 \, \text{m} \), \( \omega = 10 \, \text{rad/s} \)
\[ v_{\text{max}} = 0.05 \times 10 = 0.5 \, \text{m/s} \]

### Problem 2:

Given equation: \( x(t) = 0.1 \cos(5t + \pi/4) \)

**Amplitude**:
The amplitude \( A \) is the coefficient of the cosine function.
So, \( A = 0.1 \, \text{m} \).

**Angular Frequency**:
The angular frequency \( \omega \) is the coefficient of \( t \) in the argument of the cosine.
So, \( \omega = 5 \, \text{rad/s} \).

**Phase Constant**:
The phase constant \( \phi \) is the constant term in the argument of the cosine.
So, \( \phi = \pi/4 \).

**Period**:
\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{5} \, \text{s} \]

**Frequency**:
Frequency \( f \) is given by:
\[ f = \frac{1}{T} = \frac{5}{2\pi} \, \text{Hz} \]

These are the characteristic quantities for simple harmonic motion in the given problems.