To solve the problem, let's break down the given information and use the principles of physics, specifically the conservation of mechanical energy.
### Step 1: Determine the Initial Potential Energy
The pendulum starts at a height of 0.3 meters. The potential energy (PE) at this height is calculated using the formula:
[tex]\[ \text{PE} = m \cdot g \cdot h \][/tex]
where:
- [tex]\( m \)[/tex] is the mass (0.5 kg)
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.8 m/s²)
- [tex]\( h \)[/tex] is the height (0.3 m)
Plugging in the values, we get:
[tex]\[ \text{PE} = 0.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 0.3 \, \text{m} \][/tex]
### Step 2: Calculate the Potential Energy
[tex]\[ \text{PE} = 0.5 \times 9.8 \times 0.3 = 1.47 \, \text{J} \][/tex]
### Step 3: Convert Potential Energy to Kinetic Energy
At the lowest point of its path, all the initial potential energy will have been converted into kinetic energy (KE). The kinetic energy is given by the formula:
[tex]\[ \text{KE} = \frac{1}{2} m v^2 \][/tex]
Since the kinetic energy at the lowest point is equal to the initial potential energy:
[tex]\[ 1.47 \, \text{J} = \frac{1}{2} \times 0.5 \, \text{kg} \times v^2 \][/tex]
### Step 4: Solve for Velocity
Rearranging the equation to solve for [tex]\( v \)[/tex]:
[tex]\[ 1.47 = 0.25 \times v^2 \][/tex]
[tex]\[ v^2 = \frac{1.47}{0.25} \][/tex]
[tex]\[ v^2 = 5.88 \][/tex]
[tex]\[ v = \sqrt{5.88} \][/tex]
[tex]\[ v \approx 2.42 \, \text{m/s} \][/tex]
### Conclusion
Thus, the velocity of the pendulum when it reaches the lowest point of its path is approximately 2.42 m/s. Among the options provided, the closest answer is:
A. [tex]\( 2.4 \, \text{m/s} \)[/tex]
So, the correct answer is:
A. [tex]\( 2.4 \, \text{m/s} \)[/tex]
