To calculate the gravitational potential energy (PE) of the pendulum at the point of release, we can use the formula:
\[ PE = mgh \]
Where:
- \( m \) is the mass (0.8 kg)
- \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \))
- \( h \) is the height change from the lowest point when the pendulum is released.
To find \( h \), we need to determine how high the pendulum is raised at an angle of 50 degrees when compared to its lowest point position. The height can be found using the vertical component of the pendulum's length as follows:
- The length of the pendulum \( L \) is 1.2 m.
- The height \( h \) can be calculated as:
\[ h = L - L \cos(\theta) \]
Here, \( \theta \) is the angle in radians. First, convert \( 50 \) degrees to radians:
\[ \theta = 50 \times \left(\frac{\pi}{180}\right) \approx 0.8727 , \text{radians} \]
Now, calculate \( h \):
\[ h = 1.2 , \text{m} - 1.2 , \text{m} \cdot \cos(50^{\circ}) \]
Calculating \( \cos(50^{\circ}) \):
\[ \cos(50^{\circ}) \approx 0.6428 \]
Now, substitute this back into the height equation:
\[ h = 1.2 , \text{m} - 1.2 , \text{m} \cdot 0.6428 = 1.2 , \text{m} - 0.77136 , \text{m} \approx 0.42864 , \text{m} \]
Now that we have \( h \), we can calculate the potential energy:
\[ PE = mgh = 0.8 , \text{kg} \cdot 9.81 , \text{m/s}^2 \cdot 0.42864 , \text{m} \]
Calculating this value:
\[ PE \approx 0.8 \cdot 9.81 \cdot 0.42864 \approx 2.676 , \text{J} \]
Therefore, the gravitational potential energy of the pendulum at the point of release is approximately:
\[ \boxed{2.68 , \text{J}} \]