Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine the speed of the pendulum at the lowest point of its path, we will use the principles of conservation of energy. Specifically, we'll use the fact that the potential energy at the starting height is fully converted to kinetic energy at the lowest point.
Here’s the step-by-step solution:
1. Identify the Given Data:
- Mass of the pendulum, [tex]\( m = 1.5 \, \text{kg} \)[/tex]
- Initial height, [tex]\( h = 0.4 \, \text{m} \)[/tex]
- Acceleration due to gravity, [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex]
2. Calculate the Potential Energy at the Starting Height:
Potential energy (PE) is given by the formula:
[tex]\[ \text{PE}_{\text{initial}} = mgh \][/tex]
Substituting the values:
[tex]\[ \text{PE}_{\text{initial}} = 1.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 0.4 \, \text{m} \][/tex]
[tex]\[ \text{PE}_{\text{initial}} = 5.880 \, \text{J} \][/tex]
3. Relate Potential Energy to Kinetic Energy at the Lowest Point:
At the lowest point, all the initial potential energy is converted into kinetic energy (KE). Kinetic energy is given by the formula:
[tex]\[ \text{KE} = \frac{1}{2} mv^2 \][/tex]
Setting [tex]\(\text{PE}_{\text{initial}} = \text{KE}\)[/tex], we have:
[tex]\[ 5.880 \, \text{J} = \frac{1}{2} \times 1.5 \, \text{kg} \times v^2 \][/tex]
4. Solve for the Velocity:
[tex]\[ 5.880 = \frac{1}{2} \times 1.5 \times v^2 \][/tex]
Simplify the equation:
[tex]\[ 5.880 = 0.75 \times v^2 \][/tex]
Solve for [tex]\( v^2 \)[/tex]:
[tex]\[ v^2 = \frac{5.880}{0.75} \][/tex]
[tex]\[ v^2 = 7.840 \][/tex]
Taking the square root of both sides to find [tex]\( v \)[/tex]:
[tex]\[ v = \sqrt{7.840} \][/tex]
[tex]\[ v \approx 2.8 \, \text{m/s} \][/tex]
5. Conclusion:
The speed of the pendulum at the lowest point of its path is [tex]\( 2.8 \, \text{m/s} \)[/tex].
Therefore, the correct answer is:
A. [tex]\(2.8 \, \text{m/s}\)[/tex]
Here’s the step-by-step solution:
1. Identify the Given Data:
- Mass of the pendulum, [tex]\( m = 1.5 \, \text{kg} \)[/tex]
- Initial height, [tex]\( h = 0.4 \, \text{m} \)[/tex]
- Acceleration due to gravity, [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex]
2. Calculate the Potential Energy at the Starting Height:
Potential energy (PE) is given by the formula:
[tex]\[ \text{PE}_{\text{initial}} = mgh \][/tex]
Substituting the values:
[tex]\[ \text{PE}_{\text{initial}} = 1.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 0.4 \, \text{m} \][/tex]
[tex]\[ \text{PE}_{\text{initial}} = 5.880 \, \text{J} \][/tex]
3. Relate Potential Energy to Kinetic Energy at the Lowest Point:
At the lowest point, all the initial potential energy is converted into kinetic energy (KE). Kinetic energy is given by the formula:
[tex]\[ \text{KE} = \frac{1}{2} mv^2 \][/tex]
Setting [tex]\(\text{PE}_{\text{initial}} = \text{KE}\)[/tex], we have:
[tex]\[ 5.880 \, \text{J} = \frac{1}{2} \times 1.5 \, \text{kg} \times v^2 \][/tex]
4. Solve for the Velocity:
[tex]\[ 5.880 = \frac{1}{2} \times 1.5 \times v^2 \][/tex]
Simplify the equation:
[tex]\[ 5.880 = 0.75 \times v^2 \][/tex]
Solve for [tex]\( v^2 \)[/tex]:
[tex]\[ v^2 = \frac{5.880}{0.75} \][/tex]
[tex]\[ v^2 = 7.840 \][/tex]
Taking the square root of both sides to find [tex]\( v \)[/tex]:
[tex]\[ v = \sqrt{7.840} \][/tex]
[tex]\[ v \approx 2.8 \, \text{m/s} \][/tex]
5. Conclusion:
The speed of the pendulum at the lowest point of its path is [tex]\( 2.8 \, \text{m/s} \)[/tex].
Therefore, the correct answer is:
A. [tex]\(2.8 \, \text{m/s}\)[/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.