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 above the lowest point of the pendulum's swing.
To find \( h \), we can use the length of the pendulum and the angle it is raised:
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First, we find the vertical height \( h \):
The height \( h \) can be determined by considering the pendulum's position when raised to an angle \( \theta \) (in this case, 50 degrees).
\[ h = L - L \cos(\theta) \]
where \( L \) is the length of the pendulum (1.2 m) and \( \theta \) is the angle in radians. First, we need to convert 50 degrees to radians:
\[ \theta = 50^\circ = \frac{50 \pi}{180} \approx 0.8727 , \text{radians} \]
Now we can calculate:
\[ h = 1.2 - 1.2 \cos(50^\circ) \] Using a calculator to find \( \cos(50^\circ) \):
\[ h = 1.2 - 1.2 \cdot 0.6428 \approx 1.2 - 0.7714 \approx 0.4286 , \text{m} \]
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Now we can calculate the gravitational potential energy:
\[ PE = mgh = 0.8 \cdot 9.81 \cdot 0.4286 \]
Performing the calculation:
\[ PE \approx 0.8 \cdot 9.81 \cdot 0.4286 \approx 2.83 , \text{J} \]
It appears there was an approximation in the calculations, so let’s make sure we check against the provided options:
- \( 0.34 , \text{J} \)
- \( 2.2 , \text{J} \)
- \( 3.4 , \text{J} \)
- \( 6.0 , \text{J} \)
Given the calculation yields \( 2.83 , \text{J} \), it seems the closest option is 3.4 J.
Therefore, the pendulum's gravitational potential energy at the point of release is approximately 3.4 J.