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Let's solve the problem step by step.

### Given:
- Radius of the cylindrical well, \( r = 4 \, \text{m} \)
- Coefficient of static friction between the tyres and the wall, \( \mu = 0.4 \)
- Gravitational acceleration, \( g = 10 \, \text{m/s}^2 \)

The motorcyclist is moving in horizontal circles inside the cylindrical wall, and the minimum speed and frequency are to be calculated.

### 1. **Force Analysis:**
When the motorcyclist moves in a circle, two forces act on him:
- **Weight (mg)** acting downward.
- **Normal force (N)** exerted by the wall, acting radially inward (towards the center).

The frictional force \( f_{\text{friction}} = \mu N \) is what keeps the motorcyclist from falling due to gravity. In this case, the frictional force must balance the weight:
\[
f_{\text{friction}} = mg
\]
\[
\mu N = mg \quad \Rightarrow \quad N = \frac{mg}{\mu}
\]

### 2. **Centripetal Force:**
For circular motion, the normal force \( N \) also provides the centripetal force required to keep the motorcyclist in circular motion:
\[
N = \frac{mv^2}{r}
\]
where \( v \) is the speed of the motorcyclist, and \( r = 4 \, \text{m} \) is the radius.

### 3. **Equating the two forces:**
From the two expressions for \( N \), we have:
\[
\frac{mv^2}{r} = \frac{mg}{\mu}
\]
Simplifying:
\[
v^2 = \frac{rg}{\mu}
\]
Substitute the given values:
\[
v^2 = \frac{4 \times 10}{0.4} = 100 \quad \Rightarrow \quad v = \sqrt{100} = 10 \, \text{m/s}
\]
Thus, the **minimum speed** is \( v = 10 \, \text{m/s} \).

### 4. **Frequency:**
The frequency \( f \) is the number of revolutions per second. It is related to speed by the following relation:
\[
v = 2\pi r f
\]
Solving for \( f \):
\[
f = \frac{v}{2\pi r}
\]
Substituting the values:
\[
f = \frac{10}{2\pi \times 4} = \frac{10}{8\pi} \approx \frac{10}{25.12} \approx 0.398 \, \text{Hz}
\]
Thus, the **frequency** is approximately \( f \approx 0.4 \, \text{Hz} \).

### Final Answers:
- Minimum speed \( v = 10 \, \text{m/s} \)
- Frequency \( f \approx 0.4 \, \text{Hz} \)

Would you like further clarification on any of the steps or more details on related topics?

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Here are 5 relative questions to expand your understanding:
1. What happens to the minimum speed if the coefficient of friction is increased?
2. How would the minimum speed change if the radius of the well is doubled?
3. How does the centripetal force depend on the speed of the motorcyclist?
4. What is the role of friction in this problem, and why is static friction used instead of kinetic friction?
5. How would the required speed change if the gravitational acceleration were different (e.g., on a different planet)?

**Tip:** In circular motion problems, always identify the forces acting on the object and determine which ones provide the centripetal force to keep the object moving in a circle.

A motorcyclist (to be treated as a point mass) is to undertake horizontal circles inside the cylindrical wall of a well of inner radius 4 m. Coefficient of static friction between the tyres and the wall is 0.4. Calculate the minimum speed and frequency necessary to perform this stunt. (Use g = 10 m/s²)

Learn how to calculate the minimum speed and frequency for a motorcyclist performing horizontal circular motion inside a cylindrical wall. This problem involves concepts of circular motion, friction, and centripetal force, suitable for Grades 11-12.

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