Question

11 (a) (i) in an experiment to investigate the behaviour of a spring under a force, the following measurements were obtained.

mass hung from spring (g)	0	100	200	300	400	500	600	700	800
length of spring (mm)	60	72	84	96	108	120	132	150	180
extension (mm)

complete the table. 4

Explanation:

Step1: Recall extension formula

Extension of a spring is calculated as \( \text{extension} = \text{length of spring} - \text{original length} \). The original length (when mass is 0g) is 60 mm.

Step2: Calculate stretching force (optional, but for clarity)

Stretching force \( F = mg \), where \( g = 9.8 \, \text{m/s}^2 \) or \( 0.0098 \, \text{N/g} \). But for extension, we use original length.

Step3: Calculate extension for each mass

- For mass 0g: \( \text{extension} = 60 - 60 = 0 \, \text{mm} \)
- For mass 100g: \( 72 - 60 = 12 \, \text{mm} \)
- For mass 200g: \( 84 - 60 = 24 \, \text{mm} \)
- For mass 300g: \( 96 - 60 = 36 \, \text{mm} \)
- For mass 400g: \( 108 - 60 = 48 \, \text{mm} \)
- For mass 500g: \( 120 - 60 = 60 \, \text{mm} \)
- For mass 600g: \( 132 - 60 = 72 \, \text{mm} \)
- For mass 700g: \( 150 - 60 = 90 \, \text{mm} \)
- For mass 800g: \( 180 - 60 = 120 \, \text{mm} \)

Step4: Calculate stretching force (using \( F = mg \), \( g = 9.8 \, \text{m/s}^2 = 0.0098 \, \text{N/g} \))

- Mass 0g: \( F = 0 \times 0.0098 = 0 \, \text{N} \)
- Mass 100g: \( F = 100 \times 0.0098 = 0.98 \approx 1 \, \text{N} \)? Wait, maybe the problem has a different \( g \) or it's a simplified case. Wait, looking at the length, when mass is 0, length is 60. Let's check the stretching force. Wait, maybe the stretching force is proportional to mass. Let's see the extension and mass. For 100g, extension 12; 200g, 24 (double), so force should be proportional. Wait, maybe the stretching force is \( F = kx \), where \( x \) is extension. Let's find \( k \). For 100g, extension 12mm = 0.012m. If mass is 0.1kg, \( F = 0.1 \times 9.8 = 0.98 \, \text{N} \), so \( k = \frac{0.98}{0.012} \approx 81.67 \, \text{N/m} \). But maybe the problem expects stretching force as \( F = mg \) with \( g = 10 \, \text{N/kg} \) (simplified). So:
- Mass 0g: \( F = 0 \times 10 = 0 \, \text{N} \)
- Mass 100g = 0.1kg: \( F = 0.1 \times 10 = 1 \, \text{N} \)? But the length is 72, extension 12. Wait, maybe the stretching force is given by the extension times a constant. Wait, the table has "stretching force (N)" empty. Let's see the length and extension. Wait, maybe the stretching force is equal to the weight, so \( F = mg \), with \( g = 9.8 \, \text{N/kg} \). Let's compute:

- Mass 0g: \( F = 0 \times 9.8 = 0 \, \text{N} \)
- Mass 100g = 0.1kg: \( F = 0.1 \times 9.8 = 0.98 \approx 1 \, \text{N} \) (but maybe the problem uses \( g = 10 \) for simplicity)
- Mass 200g = 0.2kg: \( F = 0.2 \times 9.8 = 1.96 \approx 2 \, \text{N} \)
- Mass 300g = 0.3kg: \( F = 0.3 \times 9.8 = 2.94 \approx 3 \, \text{N} \)
- Mass 400g = 0.4kg: \( F = 0.4 \times 9.8 = 3.92 \approx 4 \, \text{N} \)
- Mass 500g = 0.5kg: \( F = 0.5 \times 9.8 = 4.9 \approx 5 \, \text{N} \)? Wait, no, the length at 500g is 120, extension 60. If extension is 60mm, and if \( F = kx \), then for 100g, extension 12, so \( k = F / x \). If F for 100g is 1N (simplified), then k = 1 / 0.012 ≈ 83.33 N/m. Then for 500g, extension 60mm = 0.06m, F = 83.33 * 0.06 ≈ 5N. So maybe the stretching force is 0, 1, 2, 3, 4, 5, 6, 7.5, 9 N? Wait, no, let's check the length at 700g: 150mm, extension 90mm. 90/12 = 7.5, so force would be 7.5N. At 800g: 180mm, extension 120mm, 120/12 = 10, so force 10N? Wait, maybe the problem has a typo, or maybe the stretching force is equal to the mass in grams divided by 100? Wait, 100g: 1N, 200g: 2N, ..., 500g: 5N, 600g: 6N, 700g: 7N? But extension at 700g is 90, which is 7.5*12. So maybe the stretching force is calculated as \( F = \frac{mg}{1000} \times 9.8 \), but let's focus on extension first.

The extension is length - original length (60mm). So:

| mass hung (g) | 0 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 |
|---------------|---|-----|-----|-----|-----|-----|-----|-----|-----|
| stretching force (N) | 0 | 0.98 (≈1) | 1.96 (≈2) | 2.94 (≈3) | 3.92 (≈4) | 4.9 (≈5) | 5.88 (≈6) | 6.86 (≈7) | 7.84 (≈8) | (or using g=10: 0,1,2,3,4,5,6,7,8) |
| length (mm) | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 150 | 180 |
| extension (mm) | 0 | 12 | 24 | 36 | 48 | 60 | 72 | 90 | 120 |

Wait, the extension for 700g: 150 - 60 = 90, correct. 800g: 180 - 60 = 120, correct.

So the completed table (extension column) is 0, 12, 24, 36, 48, 60, 72, 90, 120.

Stretching force: using \( F = mg \) with \( g = 9.8 \, \text{N/kg} \) (mass in kg):

- 0g: 0kg → 0N
- 100g = 0.1kg → 0.1×9.8 = 0.98N ≈1N
- 200g = 0.2kg → 0.2×9.8 = 1.96N ≈2N
- 300g = 0.3kg → 0.3×9.8 = 2.94N ≈3N
- 400g = 0.4kg → 0.4×9.8 = 3.92N ≈4N
- 500g = 0.5kg → 0.5×9.8 = 4.9N ≈5N
- 600g = 0.6kg → 0.6×9.8 = 5.88N ≈6N
- 700g = 0.7kg → 0.7×9.8 = 6.86N ≈7N (but extension is 90, which is 7.5×12, so maybe g=10? 0.7×10=7N, extension 90: 90/12=7.5, so maybe the force is 7.5N? Wait, 100g: extension 12, force 1N (12/12=1), 200g:24/12=2→2N, ..., 500g:60/12=5→5N, 600g:72/12=6→6N, 700g:90/12=7.5→7.5N, 800g:120/12=10→10N. Ah, this makes sense! So the stretching force is equal to the extension divided by 12 (since 12mm extension for 100g, so 1N per 12mm? Wait, no, extension is 12mm for 100g, so force is proportional to extension. So \( F = \frac{x}{12} \times 1 \, \text{N} \), where x is extension in mm. So:

- x=0→F=0N
- x=12→F=1N
- x=24→F=2N
- x=36→F=3N
- x=48→F=4N
- x=60→F=5N
- x=72→F=6N
- x=90→F=7.5N
- x=120→F=10N

Yes, this matches the extension and force proportionality (Hooke's law: F=kx). So the stretching force is 0, 1, 2, 3, 4, 5, 6, 7.5, 10 N.

But the problem says "Complete the table", so we need to fill stretching force and extension.

First, extension:

For each mass, extension = length - 60 (original length when mass=0).

So:

- Mass 0g: extension = 60 - 60 = 0 mm
- Mass 100g: 72 - 60 = 12 mm
- Mass 200g: 84 - 60 = 24 mm
- Mass 300g: 96 - 60 = 36 mm
- Mass 400g: 108 - 60 = 48 mm
- Mass 500g: 120 - 60 = 60 mm
- Mass 600g: 132 - 60 = 72 mm
- Mass 700g: 150 - 60 = 90 mm
- Mass 800g: 180 - 60 = 120 mm

Stretching force: Using Hooke's law, F = kx. From mass 100g, x=12mm, F should be the weight, which is mg. If mass=100g=0.1kg, g=9.8N/kg, F=0.98N≈1N. So k = F/x = 0.98N / 0.012m ≈81.67N/m. Then for x=12mm=0.012m, F=81.67×0.012≈0.98N≈1N. So the stretching force is approximately 0, 1, 2, 3, 4, 5, 6, 7, 8 N (using g=10 for simplicity: 0,1,2,3,4,5,6,7,8 N) or more accurately 0, 0.98, 1.96, 2.94, 3.92, 4.9, 5.88, 6.86, 7.84 N. But since the extension is 12,24,...120, which is 12×1, 12×2,...12×10, the force is 1,2,...10 N if we take g=10 and mass in 100g units (100g=0.1kg, 0.1×10=1N, so 100g→1N, 200g→2N, ..., 800g→8N? But extension at 800g is 120, which is 12×10, so 10N. So there's a discrepancy. Wait, maybe the problem has a mistake, but the key is to calculate extension as length - original length (60mm), and stretching force as mg (mass in kg, g=9.8 or 10).

But the main task is to complete the table. So:

Stretching Force (N) Calculation:

Using \( F = mg \), where \( m \) is mass in kg, \( g = 9.8 \, \text{N/kg} \):

- \( m = 0 \, \text{kg} \): \( F = 0 \times 9.8 = 0 \, \text{N} \)
- \( m = 0.1 \, \text{kg} \) (100g): \( F = 0.1 \times 9.8 = 0.98 \, \text{N} \approx 1 \, \text{N} \)
- \( m = 0.2 \, \text{kg} \) (200g): \( F = 0.2 \times 9.8 = 1.96 \, \text{N} \approx 2 \, \text{N} \)
- \( m = 0.3 \, \text{kg} \) (300g): \( F = 0.3 \times 9.8 = 2.94 \, \text{N} \approx 3 \, \text{N} \)
- \( m = 0.4 \, \text{kg} \) (400g): \( F = 0.4 \times 9.8 = 3.92 \, \text{N} \approx 4 \, \text{N} \)
- \( m = 0.5 \, \text{kg} \) (500g): \( F = 0.5 \times 9.8 = 4.9 \, \text{N} \approx 5 \, \text{N} \)
- \( m = 0.6 \, \text{kg} \) (600g): \( F = 0.6 \times 9.8 = 5.88 \, \text{N} \approx 6 \, \text{N} \)
- \( m = 0.7 \, \text{kg} \) (700g): \( F = 0.7 \times 9.8 = 6.86 \, \text{N} \approx 7 \, \text{N} \)
- \( m = 0.8 \, \text{kg} \) (800g): \( F = 0.8 \times 9.8 = 7.84 \, \text{N} \approx 8 \, \text{N} \)

Extension (mm) Calculation:

\( \text{extension} = \text{length} - 60 \):

- Length 60mm: \( 60 - 60 = 0 \, \text{mm} \)
-

Answer:

Step1: Recall extension formula

Extension of a spring is calculated as \( \text{extension} = \text{length of spring} - \text{original length} \). The original length (when mass is 0g) is 60 mm.

Step2: Calculate stretching force (optional, but for clarity)

Stretching force \( F = mg \), where \( g = 9.8 \, \text{m/s}^2 \) or \( 0.0098 \, \text{N/g} \). But for extension, we use original length.

Step3: Calculate extension for each mass

• For mass 0g: \( \text{extension} = 60 - 60 = 0 \, \text{mm} \)
• For mass 100g: \( 72 - 60 = 12 \, \text{mm} \)
• For mass 200g: \( 84 - 60 = 24 \, \text{mm} \)
• For mass 300g: \( 96 - 60 = 36 \, \text{mm} \)
• For mass 400g: \( 108 - 60 = 48 \, \text{mm} \)
• For mass 500g: \( 120 - 60 = 60 \, \text{mm} \)
• For mass 600g: \( 132 - 60 = 72 \, \text{mm} \)
• For mass 700g: \( 150 - 60 = 90 \, \text{mm} \)
• For mass 800g: \( 180 - 60 = 120 \, \text{mm} \)

Step4: Calculate stretching force (using \( F = mg \), \( g = 9.8 \, \text{m/s}^2 = 0.0098 \, \text{N/g} \))

• Mass 0g: \( F = 0 \times 0.0098 = 0 \, \text{N} \)
• Mass 100g: \( F = 100 \times 0.0098 = 0.98 \approx 1 \, \text{N} \)? Wait, maybe the problem has a different \( g \) or it's a simplified case. Wait, looking at the length, when mass is 0, length is 60. Let's check the stretching force. Wait, maybe the stretching force is proportional to mass. Let's see the extension and mass. For 100g, extension 12; 200g, 24 (double), so force should be proportional. Wait, maybe the stretching force is \( F = kx \), where \( x \) is extension. Let's find \( k \). For 100g, extension 12mm = 0.012m. If mass is 0.1kg, \( F = 0.1 \times 9.8 = 0.98 \, \text{N} \), so \( k = \frac{0.98}{0.012} \approx 81.67 \, \text{N/m} \). But maybe the problem expects stretching force as \( F = mg \) with \( g = 10 \, \text{N/kg} \) (simplified). So:
• Mass 0g: \( F = 0 \times 10 = 0 \, \text{N} \)
• Mass 100g = 0.1kg: \( F = 0.1 \times 10 = 1 \, \text{N} \)? But the length is 72, extension 12. Wait, maybe the stretching force is given by the extension times a constant. Wait, the table has "stretching force (N)" empty. Let's see the length and extension. Wait, maybe the stretching force is equal to the weight, so \( F = mg \), with \( g = 9.8 \, \text{N/kg} \). Let's compute:

• Mass 0g: \( F = 0 \times 9.8 = 0 \, \text{N} \)
• Mass 100g = 0.1kg: \( F = 0.1 \times 9.8 = 0.98 \approx 1 \, \text{N} \) (but maybe the problem uses \( g = 10 \) for simplicity)
• Mass 200g = 0.2kg: \( F = 0.2 \times 9.8 = 1.96 \approx 2 \, \text{N} \)
• Mass 300g = 0.3kg: \( F = 0.3 \times 9.8 = 2.94 \approx 3 \, \text{N} \)
• Mass 400g = 0.4kg: \( F = 0.4 \times 9.8 = 3.92 \approx 4 \, \text{N} \)
• Mass 500g = 0.5kg: \( F = 0.5 \times 9.8 = 4.9 \approx 5 \, \text{N} \)? Wait, no, the length at 500g is 120, extension 60. If extension is 60mm, and if \( F = kx \), then for 100g, extension 12, so \( k = F / x \). If F for 100g is 1N (simplified), then k = 1 / 0.012 ≈ 83.33 N/m. Then for 500g, extension 60mm = 0.06m, F = 83.33 0.06 ≈ 5N. So maybe the stretching force is 0, 1, 2, 3, 4, 5, 6, 7.5, 9 N? Wait, no, let's check the length at 700g: 150mm, extension 90mm. 90/12 = 7.5, so force would be 7.5N. At 800g: 180mm, extension 120mm, 120/12 = 10, so force 10N? Wait, maybe the problem has a typo, or maybe the stretching force is equal to the mass in grams divided by 100? Wait, 100g: 1N, 200g: 2N, ..., 500g: 5N, 600g: 6N, 700g: 7N? But extension at 700g is 90, which is 7.512. So maybe the stretching force is calculated as \( F = \frac{mg}{1000} \times 9.8 \), but let's focus on extension first.

The extension is length - original length (60mm). So:

mass hung (g)	0	100	200	300	400	500	600	700	800
length (mm)	60	72	84	96	108	120	132	150	180
extension (mm)	0	12	24	36	48	60	72	90	120

Wait, the extension for 700g: 150 - 60 = 90, correct. 800g: 180 - 60 = 120, correct.

So the completed table (extension column) is 0, 12, 24, 36, 48, 60, 72, 90, 120.

Stretching force: using \( F = mg \) with \( g = 9.8 \, \text{N/kg} \) (mass in kg):

• 0g: 0kg → 0N
• 100g = 0.1kg → 0.1×9.8 = 0.98N ≈1N
• 200g = 0.2kg → 0.2×9.8 = 1.96N ≈2N
• 300g = 0.3kg → 0.3×9.8 = 2.94N ≈3N
• 400g = 0.4kg → 0.4×9.8 = 3.92N ≈4N
• 500g = 0.5kg → 0.5×9.8 = 4.9N ≈5N
• 600g = 0.6kg → 0.6×9.8 = 5.88N ≈6N
• 700g = 0.7kg → 0.7×9.8 = 6.86N ≈7N (but extension is 90, which is 7.5×12, so maybe g=10? 0.7×10=7N, extension 90: 90/12=7.5, so maybe the force is 7.5N? Wait, 100g: extension 12, force 1N (12/12=1), 200g:24/12=2→2N, ..., 500g:60/12=5→5N, 600g:72/12=6→6N, 700g:90/12=7.5→7.5N, 800g:120/12=10→10N. Ah, this makes sense! So the stretching force is equal to the extension divided by 12 (since 12mm extension for 100g, so 1N per 12mm? Wait, no, extension is 12mm for 100g, so force is proportional to extension. So \( F = \frac{x}{12} \times 1 \, \text{N} \), where x is extension in mm. So:

• x=0→F=0N
• x=12→F=1N
• x=24→F=2N
• x=36→F=3N
• x=48→F=4N
• x=60→F=5N
• x=72→F=6N
• x=90→F=7.5N
• x=120→F=10N

Yes, this matches the extension and force proportionality (Hooke's law: F=kx). So the stretching force is 0, 1, 2, 3, 4, 5, 6, 7.5, 10 N.

But the problem says "Complete the table", so we need to fill stretching force and extension.

First, extension:

For each mass, extension = length - 60 (original length when mass=0).

So:

• Mass 0g: extension = 60 - 60 = 0 mm
• Mass 100g: 72 - 60 = 12 mm
• Mass 200g: 84 - 60 = 24 mm
• Mass 300g: 96 - 60 = 36 mm
• Mass 400g: 108 - 60 = 48 mm
• Mass 500g: 120 - 60 = 60 mm
• Mass 600g: 132 - 60 = 72 mm
• Mass 700g: 150 - 60 = 90 mm
• Mass 800g: 180 - 60 = 120 mm

Stretching force: Using Hooke's law, F = kx. From mass 100g, x=12mm, F should be the weight, which is mg. If mass=100g=0.1kg, g=9.8N/kg, F=0.98N≈1N. So k = F/x = 0.98N / 0.012m ≈81.67N/m. Then for x=12mm=0.012m, F=81.67×0.012≈0.98N≈1N. So the stretching force is approximately 0, 1, 2, 3, 4, 5, 6, 7, 8 N (using g=10 for simplicity: 0,1,2,3,4,5,6,7,8 N) or more accurately 0, 0.98, 1.96, 2.94, 3.92, 4.9, 5.88, 6.86, 7.84 N. But since the extension is 12,24,...120, which is 12×1, 12×2,...12×10, the force is 1,2,...10 N if we take g=10 and mass in 100g units (100g=0.1kg, 0.1×10=1N, so 100g→1N, 200g→2N, ..., 800g→8N? But extension at 800g is 120, which is 12×10, so 10N. So there's a discrepancy. Wait, maybe the problem has a mistake, but the key is to calculate extension as length - original length (60mm), and stretching force as mg (mass in kg, g=9.8 or 10).

But the main task is to complete the table. So:

Stretching Force (N) Calculation:

Using \( F = mg \), where \( m \) is mass in kg, \( g = 9.8 \, \text{N/kg} \):

• \( m = 0 \, \text{kg} \): \( F = 0 \times 9.8 = 0 \, \text{N} \)
• \( m = 0.1 \, \text{kg} \) (100g): \( F = 0.1 \times 9.8 = 0.98 \, \text{N} \approx 1 \, \text{N} \)
• \( m = 0.2 \, \text{kg} \) (200g): \( F = 0.2 \times 9.8 = 1.96 \, \text{N} \approx 2 \, \text{N} \)
• \( m = 0.3 \, \text{kg} \) (300g): \( F = 0.3 \times 9.8 = 2.94 \, \text{N} \approx 3 \, \text{N} \)
• \( m = 0.4 \, \text{kg} \) (400g): \( F = 0.4 \times 9.8 = 3.92 \, \text{N} \approx 4 \, \text{N} \)
• \( m = 0.5 \, \text{kg} \) (500g): \( F = 0.5 \times 9.8 = 4.9 \, \text{N} \approx 5 \, \text{N} \)
• \( m = 0.6 \, \text{kg} \) (600g): \( F = 0.6 \times 9.8 = 5.88 \, \text{N} \approx 6 \, \text{N} \)
• \( m = 0.7 \, \text{kg} \) (700g): \( F = 0.7 \times 9.8 = 6.86 \, \text{N} \approx 7 \, \text{N} \)
• \( m = 0.8 \, \text{kg} \) (800g): \( F = 0.8 \times 9.8 = 7.84 \, \text{N} \approx 8 \, \text{N} \)

Extension (mm) Calculation:

\( \text{extension} = \text{length} - 60 \):

• Length 60mm: \( 60 - 60 = 0 \, \text{mm} \)

-