Question

read each scenario and then answer the question.
scenario a: a 3 \\(\frac{n}{m}\\) spring is compressed a distance of 1.0 m.
scenario b: a 6 \\(\frac{n}{m}\\) spring is compressed a distance of 0.8 m.
scenario c: a 9 \\(\frac{n}{m}\\) spring is compressed a distance of 0.6 m.
scenario d: a 12 \\(\frac{n}{m}\\) spring is compressed a distance of 0.4 m.

which scenario generates the most elastic potential energy?
options: a, b, c, d

Explanation:

Step1: Recall the formula for elastic potential energy

The formula for elastic potential energy (EPE) of a spring is $EPE = \frac{1}{2}kx^2$, where $k$ is the spring constant (in $\frac{N}{m}$) and $x$ is the compression or extension distance (in $m$).

Step2: Calculate EPE for Scenario A

For Scenario A: $k = 3\frac{N}{m}$, $x = 1.0\ m$
$EPE_A = \frac{1}{2} \times 3 \times (1.0)^2 = \frac{1}{2} \times 3 \times 1 = 1.5\ J$

Step3: Calculate EPE for Scenario B

For Scenario B: $k = 6\frac{N}{m}$, $x = 0.8\ m$
$EPE_B = \frac{1}{2} \times 6 \times (0.8)^2 = 3 \times 0.64 = 1.92\ J$

Step4: Calculate EPE for Scenario C

For Scenario C: $k = 9\frac{N}{m}$, $x = 0.6\ m$
$EPE_C = \frac{1}{2} \times 9 \times (0.6)^2 = 4.5 \times 0.36 = 1.62\ J$

Step5: Calculate EPE for Scenario D

For Scenario D: $k = 12\frac{N}{m}$, $x = 0.4\ m$
$EPE_D = \frac{1}{2} \times 12 \times (0.4)^2 = 6 \times 0.16 = 0.96\ J$

Step6: Compare the EPE values

Compare $EPE_A = 1.5\ J$, $EPE_B = 1.92\ J$, $EPE_C = 1.62\ J$, $EPE_D = 0.96\ J$.
We see that $EPE_B$ (1.92 J) is greater than $EPE_A$, $EPE_C$, and $EPE_D$. Wait, wait, did I miscalculate? Wait, no, let's recheck Scenario B: $k=6$, $x=0.8$. $\frac{1}{2}*6*(0.8)^2 = 3*0.64 = 1.92$. Scenario C: $\frac{1}{2}*9*(0.6)^2 = 4.5*0.36 = 1.62$. Scenario D: $\frac{1}{2}*12*(0.4)^2 = 6*0.16 = 0.96$. Scenario A: 1.5. Wait, but wait, maybe I made a mistake. Wait, no, let's check again. Wait, Scenario B: 6 N/m, 0.8 m. So 0.5*6*0.8² = 3*0.64=1.92. Scenario C: 0.5*9*0.6²=4.5*0.36=1.62. Scenario D: 0.5*12*0.4²=6*0.16=0.96. Scenario A: 0.5*3*1²=1.5. Wait, but wait, maybe I messed up. Wait, no, the correct calculation: Wait, Scenario B: 6*0.8²/2 = 6*0.64/2=3.84/2=1.92? Wait, no! Wait, $\frac{1}{2}kx^2$: for k=6, x=0.8: 0.5*6*0.64=3*0.64=1.92. Wait, no, 6*0.64 is 3.84, divided by 2 is 1.92. Yes. Scenario C: 9*0.36=3.24, divided by 2 is 1.62. Scenario D: 12*0.16=1.92, divided by 2 is 0.96. Scenario A: 3*1=3, divided by 2 is 1.5. Wait, no, wait, 0.5*3*1² is 1.5. Wait, but wait, Scenario B: 0.5*6*0.8²=0.5*6*0.64=1.92. Scenario C: 0.5*9*0.6²=0.5*9*0.36=1.62. Scenario D: 0.5*12*0.4²=0.5*12*0.16=0.96. Scenario A: 1.5. Wait, but wait, maybe I made a mistake in Scenario B. Wait, no, let's recalculate Scenario B: $k=6$, $x=0.8$. $x^2=0.64$. $0.5*6=3$. $3*0.64=1.92$. Correct. Scenario C: $k=9$, $x=0.6$. $x^2=0.36$. $0.5*9=4.5$. $4.5*0.36=1.62$. Correct. Scenario D: $k=12$, $x=0.4$. $x^2=0.16$. $0.5*12=6$. $6*0.16=0.96$. Correct. Scenario A: $k=3$, $x=1.0$. $x^2=1$. $0.5*3=1.5$. $1.5*1=1.5$. So Scenario B has 1.92 J, which is higher than A (1.5), C (1.62), D (0.96). Wait, but wait, Scenario C: 9*0.6²/2 = 9*0.36/2=3.24/2=1.62. Scenario B: 6*0.8²/2=6*0.64/2=3.84/2=1.92. So B is higher than C. Wait, but the options are A, B, C, D. So the answer is B? Wait, but let me check again. Wait, maybe I miscalculated Scenario C. Wait, 9 N/m, 0.6 m. So EPE = 0.5*9*0.6² = 0.5*9*0.36 = 4.5*0.36 = 1.62. Correct. Scenario B: 0.5*6*0.8² = 3*0.64 = 1.92. Correct. So B has more than C. So the answer is B? Wait, but the user's question is which scenario generates the most elastic potential energy. So according to the calculations, Scenario B has 1.92 J, Scenario A has 1.5 J, Scenario C has 1.62 J, Scenario D has 0.96 J. So Scenario B has the most? Wait, no, wait, wait, I think I made a mistake in Scenario B's calculation. Wait, 0.8 squared is 0.64, multiplied by 6 is 3.84, divided by 2 is 1.92. Correct. Scenario C: 0.6 squared is 0.36, multiplied by 9 is 3.24, divided by 2 is 1.62. Correct. So yes, Scenario B has the highest. Wait, but let me check the problem again. The spring constants are 3, 6, 9, 12 N/m, and compressions 1.0, 0.8, 0.6, 0.4 m. So using EPE = 0.5*k*x². So:

A: 0.5*3*1² = 1.5 J

B: 0.5*6*0.8² = 0.5*6*0.64 = 1.92 J

C: 0.5*9*0.6² = 0.5*9*0.36 = 1.62 J

D: 0.5*12*0.4² = 0.5*12*0.16 = 0.96 J

So Scenario B has the most elastic potential energy.

Answer:

B. Scenario B