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To determine which scenario generates the most elastic potential energy, we need to calculate the elastic potential energy [tex]\( U \)[/tex] for each of the four scenarios. The formula for elastic potential energy in a spring is given by:
[tex]\[ U = \frac{1}{2} k x^2 \][/tex]
where [tex]\( k \)[/tex] is the spring constant and [tex]\( x \)[/tex] is the displacement (or compression) from the equilibrium position.
Let's calculate the elastic potential energy for each scenario:
### Scenario A
- Spring constant [tex]\( k_A = 3 \frac{N}{m} \)[/tex]
- Compression [tex]\( x_A = 1.0 \, m \)[/tex]
[tex]\[ U_A = \frac{1}{2} k_A x_A^2 \\ U_A = \frac{1}{2} \times 3 \times (1.0)^2 \\ U_A = \frac{1}{2} \times 3 \times 1.0 \\ U_A = \frac{3}{2} \\ U_A = 1.5 \, \text{J} \][/tex]
### Scenario B
- Spring constant [tex]\( k_B = 6 \frac{N}{m} \)[/tex]
- Compression [tex]\( x_B = 0.8 \, m \)[/tex]
[tex]\[ U_B = \frac{1}{2} k_B x_B^2 \\ U_B = \frac{1}{2} \times 6 \times (0.8)^2 \\ U_B = \frac{1}{2} \times 6 \times 0.64 \\ U_B = \frac{6 \times 0.64}{2} \\ U_B = 1.920 \, \text{J} \][/tex]
### Scenario C
- Spring constant [tex]\( k_C = 9 \frac{N}{m} \)[/tex]
- Compression [tex]\( x_C = 0.6 \, m \)[/tex]
[tex]\[ U_C = \frac{1}{2} k_C x_C^2 \\ U_C = \frac{1}{2} \times 9 \times (0.6)^2 \\ U_C = \frac{1}{2} \times 9 \times 0.36 \\ U_C = \frac{9 \times 0.36}{2} \\ U_C = 1.620 \, \text{J} \][/tex]
### Scenario D
- Spring constant [tex]\( k_D = 12 \frac{N}{m} \)[/tex]
- Compression [tex]\( x_D = 0.4 \, m \)[/tex]
[tex]\[ U_D = \frac{1}{2} k_D x_D^2 \\ U_D = \frac{1}{2} \times 12 \times (0.4)^2 \\ U_D = \frac{1}{2} \times 12 \times 0.16 \\ U_D = \frac{12 \times 0.16}{2} \\ U_D = 0.960 \, \text{J} \][/tex]
### Comparison
Now we have the calculated elastic potential energies for each scenario:
- [tex]\( U_A = 1.5 \, \text{J} \)[/tex]
- [tex]\( U_B = 1.920 \, \text{J} \)[/tex]
- [tex]\( U_C = 1.620 \, \text{J} \)[/tex]
- [tex]\( U_D = 0.960 \, \text{J} \)[/tex]
The highest value among these elastic potential energies is [tex]\( 1.920 \, \text{J} \)[/tex], which comes from Scenario B.
Therefore, the scenario that generates the most elastic potential energy is:
Scenario B.
[tex]\[ U = \frac{1}{2} k x^2 \][/tex]
where [tex]\( k \)[/tex] is the spring constant and [tex]\( x \)[/tex] is the displacement (or compression) from the equilibrium position.
Let's calculate the elastic potential energy for each scenario:
### Scenario A
- Spring constant [tex]\( k_A = 3 \frac{N}{m} \)[/tex]
- Compression [tex]\( x_A = 1.0 \, m \)[/tex]
[tex]\[ U_A = \frac{1}{2} k_A x_A^2 \\ U_A = \frac{1}{2} \times 3 \times (1.0)^2 \\ U_A = \frac{1}{2} \times 3 \times 1.0 \\ U_A = \frac{3}{2} \\ U_A = 1.5 \, \text{J} \][/tex]
### Scenario B
- Spring constant [tex]\( k_B = 6 \frac{N}{m} \)[/tex]
- Compression [tex]\( x_B = 0.8 \, m \)[/tex]
[tex]\[ U_B = \frac{1}{2} k_B x_B^2 \\ U_B = \frac{1}{2} \times 6 \times (0.8)^2 \\ U_B = \frac{1}{2} \times 6 \times 0.64 \\ U_B = \frac{6 \times 0.64}{2} \\ U_B = 1.920 \, \text{J} \][/tex]
### Scenario C
- Spring constant [tex]\( k_C = 9 \frac{N}{m} \)[/tex]
- Compression [tex]\( x_C = 0.6 \, m \)[/tex]
[tex]\[ U_C = \frac{1}{2} k_C x_C^2 \\ U_C = \frac{1}{2} \times 9 \times (0.6)^2 \\ U_C = \frac{1}{2} \times 9 \times 0.36 \\ U_C = \frac{9 \times 0.36}{2} \\ U_C = 1.620 \, \text{J} \][/tex]
### Scenario D
- Spring constant [tex]\( k_D = 12 \frac{N}{m} \)[/tex]
- Compression [tex]\( x_D = 0.4 \, m \)[/tex]
[tex]\[ U_D = \frac{1}{2} k_D x_D^2 \\ U_D = \frac{1}{2} \times 12 \times (0.4)^2 \\ U_D = \frac{1}{2} \times 12 \times 0.16 \\ U_D = \frac{12 \times 0.16}{2} \\ U_D = 0.960 \, \text{J} \][/tex]
### Comparison
Now we have the calculated elastic potential energies for each scenario:
- [tex]\( U_A = 1.5 \, \text{J} \)[/tex]
- [tex]\( U_B = 1.920 \, \text{J} \)[/tex]
- [tex]\( U_C = 1.620 \, \text{J} \)[/tex]
- [tex]\( U_D = 0.960 \, \text{J} \)[/tex]
The highest value among these elastic potential energies is [tex]\( 1.920 \, \text{J} \)[/tex], which comes from Scenario B.
Therefore, the scenario that generates the most elastic potential energy is:
Scenario B.
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