Question: 1. The coefficient of friction, $\mu$, can be determined in an experiment by measuring the force $F$ required to move a mass $m$. When $F$ is measured and $m$ is known, the coefficient of friction can by calculated by:
$$\mu = \frac{F}{mg}$$
Let $g = 9.81$ ms$^{-2}$. Results from measuring $F$ in six tests are given in the table below. Determine the coefficient of friction in each test, and the average from all the tests (2 outputs, make sure to comment and label appropriately).
Test 1 2 3 4 5 6
Mass $m$ (kg) 2 4 5 10 20 50
Force $F$ (N) 12.5 23.5 30 61 117 294
Top-left: Text paragraph with the formula $\mu = \frac{F}{mg}$ near the upper left, and a table below it listing six tests with mass and force values. What is the answer for this?
1. **State the problem:** We need to calculate the coefficient of friction $\mu$ for each test using the formula $$\mu = \frac{F}{mg}$$ where $F$ is the force, $m$ is the mass, and $g = 9.81$ m/s$^2$ is the acceleration due to gravity.
2. **Given data:**
| Test | 1 | 2 | 3 | 4 | 5 | 6 |
|-------|---|---|---|----|----|----|
| Mass $m$ (kg) | 2 | 4 | 5 | 10 | 20 | 50 |
| Force $F$ (N) | 12.5 | 23.5 | 30 | 61 | 117 | 294 |
3. **Calculate $\mu$ for each test:**
$$\mu = \frac{F}{mg}$$
Calculate denominator $mg$ for each test:
- Test 1: $2 \times 9.81 = 19.62$
- Test 2: $4 \times 9.81 = 39.24$
- Test 3: $5 \times 9.81 = 49.05$
- Test 4: $10 \times 9.81 = 98.1$
- Test 5: $20 \times 9.81 = 196.2$
- Test 6: $50 \times 9.81 = 490.5$
Now calculate $\mu$:
- Test 1: $$\mu = \frac{12.5}{19.62} = 0.637$$
- Test 2: $$\mu = \frac{23.5}{39.24} = 0.599$$
- Test 3: $$\mu = \frac{30}{49.05} = 0.611$$
- Test 4: $$\mu = \frac{61}{98.1} = 0.622$$
- Test 5: $$\mu = \frac{117}{196.2} = 0.596$$
- Test 6: $$\mu = \frac{294}{490.5} = 0.599$$
4. **Calculate the average coefficient of friction:**
$$\text{Average } \mu = \frac{0.637 + 0.599 + 0.611 + 0.622 + 0.596 + 0.599}{6}$$
$$= \frac{3.664}{6} = 0.611$$
5. **Final answers:**
- Coefficients of friction for each test:
- Test 1: $0.637$
- Test 2: $0.599$
- Test 3: $0.611$
- Test 4: $0.622$
- Test 5: $0.596$
- Test 6: $0.599$
- Average coefficient of friction: $0.611$
These values indicate the frictional force relative to the normal force for each test.