To calculate the gravitational potential energy (GPE) of a pendulum at the point of release, we can use the following formula:
\[ \text{GPE} = mgh \]
where:
- \( m \) is the mass (in kg),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)),
- \( h \) is the height above the lowest point of the swing.
First, we need to find the height \( h \) when the pendulum is raised to an angle of 50 degrees. The height can be derived from the length of the pendulum and the angle it is raised:
- The vertical height \( h \) can be calculated using the formula:
\[ h = L - L \cdot \cos(\theta) \]
where \( L \) is the length of the pendulum and \( \theta \) is the angle from the vertical.
Given:
- \( L = 1.2 , \text{m} \)
- \( \theta = 50^\circ \)
First, we convert the angle to radians for calculation purposes, or keep it in degrees if your calculator can handle it. For simplicity, we'll use degrees.
Now, calculate \( h \):
\[ h = 1.2 - 1.2 \cdot \cos(50^\circ) \]
Using \( \cos(50^\circ) \approx 0.6428 \):
\[ h \approx 1.2 - 1.2 \cdot 0.6428 \approx 1.2 - 0.7714 \approx 0.4286 , \text{m} \]
Now, substitute \( m \), \( g \), and \( h \) into the GPE formula:
\[ \text{GPE} = 0.8 , \text{kg} \cdot 9.81 , \text{m/s}^2 \cdot 0.4286 , \text{m} \]
Calculating this gives:
\[ \text{GPE} \approx 0.8 \cdot 9.81 \cdot 0.4286 \approx 2.83 , \text{J} \]
Thus, the gravitational potential energy at the point of release is approximately 2.83 Joules.