In this exercise, we explore the dynamics of a pendulum to determine its maximum height based on given parameters. We start with a pendulum of length \( L = 2 \) meters and a mass \( m = 0.4 \) kg, which is at a 5-degree angle from the vertical while moving at a speed of \( v = 1.5 \) m/s. The goal is to find the maximum height \( h_{\text{max}} \) that the pendulum reaches during its swing.

To solve this, we apply the principle of conservation of mechanical energy, which states that the total mechanical energy in a closed system remains constant. The relevant equation for energy conservation in this context is:

\[ mgh_{\text{max}} = mgh_p + \frac{1}{2} mv^2 \]

Here, \( h_p \) is the height of the pendulum at the specific point where it makes a 5-degree angle. To find \( h_p \), we use the formula:

\[ h_p = L(1 - \cos(\theta_p)) \]

Before substituting values, we need to convert the angle from degrees to radians:

\[ \theta_p = 5 \times \frac{\pi}{180} \approx 0.087 \text{ radians} \]

Now, substituting the values into the height equation:

\[ h_p = 2(1 - \cos(0.087)) \approx 2(1 - 0.9962) \approx 0.0076 \text{ meters} \]

With \( h_p \) calculated, we can now substitute back into the energy conservation equation. Rearranging gives us:

\[ h_{\text{max}} = h_p + \frac{v^2}{2g} \]

Substituting the known values, where \( g \approx 9.8 \, \text{m/s}^2 \):

\[ h_{\text{max}} = 0.0076 + \frac{(1.5)^2}{2 \times 9.8} \]

Calculating the kinetic energy term:

\[ \frac{(1.5)^2}{2 \times 9.8} = \frac{2.25}{19.6} \approx 0.1153 \]

Now, adding this to \( h_p \):

\[ h_{\text{max}} = 0.0076 + 0.1153 \approx 0.1229 \text{ meters} \]

Thus, the maximum height \( h_{\text{max}} \) that the pendulum reaches is approximately \( 0.12 \) meters. This exercise illustrates the application of energy conservation principles in analyzing pendulum motion, emphasizing the relationship between potential and kinetic energy at different points in the swing.

Struggling with Physics?