Which is true for rolling friction `( mu _(r ) ) `, static friction `( mu_(s))` and kinetic friction `( mu_(k))` ?

Options

A. (a)`mu_(s) gt mu_(k) gt mu_(r)`
B. (b)`mu_(s) lt mu_(k) lt mu_(r)`
C. (c)`mu_(s) lt mu_(k) gt mu_(r)`
D. (d)`mu_(s) gt mu_(k) gt mu_(r)`

Answer

To solve the question regarding the relationship between rolling friction (μ_r), static friction (μ_s), and kinetic friction (μ_k), let's analyze the properties of these types of friction step by step.

### Step-by-Step Solution:

1. **Understanding Friction Types**:
- **Static Friction (μ_s)**: This is the friction that prevents an object from starting to move. It acts when a force is applied to an object at rest.
- **Kinetic Friction (μ_k)**: This is the friction that acts on an object that is already in motion. It is usually less than static friction.
- **Rolling Friction (μ_r)**: This is the friction that occurs when an object rolls over a surface. It is generally the least among the three types of friction.

2. **Comparing the Magnitudes**:
- It is known from physics that:
- The maximum static friction (μ_s) is greater than kinetic friction (μ_k) when an object is in motion.
- Kinetic friction (μ_k) is greater than rolling friction (μ_r) when an object rolls.
- Therefore, we can establish the following inequality:
\[ μ_s > μ_k > μ_r \]

3. **Conclusion**:
- Based on the above comparison, we can conclude that the relationship among the coefficients of friction is:
\[ μ_s > μ_k > μ_r \]
- This means that static friction is the greatest, followed by kinetic friction, and rolling friction is the least.

### Final Answer:
The correct statement is: **μ_s > μ_k > μ_r**.