To determine the fraction of the chain's length that is on the smooth surface, we need to analyze the forces acting on the chain and apply the given conditions appropriately.
We are given:
- A horizontal force [tex]\( F \)[/tex] producing a unit acceleration.
- The mass of the chain [tex]\( m \)[/tex].
- The acceleration due to gravity [tex]\( g \)[/tex].
- The coefficient of friction [tex]\( \mu \)[/tex].
### Step-by-Step Solution
#### 1. Understanding the Forces
The chain is subjected to two different types of surface:
- A smooth surface (no friction).
- A rough surface (with friction).
Given the coefficient of friction [tex]\(\mu\)[/tex] on the rough surface, the frictional force [tex]\( F_{\text{friction}} \)[/tex] can be calculated as:
[tex]\[ F_{\text{friction}} = \mu \times \text{(normal force)} = \mu \times (m_{\text{rough}} \times g), \][/tex]
where [tex]\( m_{\text{rough}} \)[/tex] is the mass of the chain on the rough surface.
Since we are dealing with horizontal surfaces:
[tex]\[ m_{\text{rough}} g \approx w_{\text{rough}},\][/tex]
where [tex]\( w_{\text{rough}} \)[/tex] is the weight of the chain on the rough surface.
#### 2. Applying Newton's Second Law
The total force [tex]\( F \)[/tex] is used to overcome the frictional force and accelerate the mass with unit acceleration. Hence, we can write:
[tex]\[ F - F_{\text{friction}} = m \times 1 \][/tex] (since unit acceleration implies [tex]\(a = 1\)[/tex]).
Replacing [tex]\( F_{\text{friction}} \)[/tex],
[tex]\[ F - \mu \times w_{\text{rough}} = m \times 1. \][/tex]
[tex]\[ F - \mu (m_{\text{rough}} \times g) = m. \][/tex]
#### 3. Simplifying the Expression
The fraction of the chain on the smooth surface is given by [tex]\( \frac{w_{\text{smooth}}}{w_{\text{total}}} \)[/tex], where [tex]\( w_{\text{total}} = m \times g \)[/tex], and [tex]\( m_{\text{smooth}} = w_{\text{smooth}} / g \)[/tex].
By substituting [tex]\( m_{\text{rough}} = m - m_{\text{smooth}} \)[/tex]:
[tex]\[ F - \mu \left( (m - w_{\text{smooth}}/g) \times g \right) = m. \][/tex]
[tex]\[ F - \mu (m g - w_{\text{smooth}}) = m. \][/tex]
Solving for [tex]\( w_{\text{smooth}} \)[/tex]:
[tex]\[ F - \mu mg + \mu w_{\text{smooth}} = m. \][/tex]
[tex]\[ \mu w_{\text{smooth}} = m + \mu mg - F. \][/tex]
[tex]\[ w_{\text{smooth}} = \frac{ m + \mu mg - F }{ \mu }. \][/tex]
Expressing in terms of the length fraction, [tex]\( w_{\text{smooth}} \)[/tex]:
[tex]\[ \text{Fraction length smooth} = \frac{w_{\text{smooth}}}{w_{\text{total}}}. \][/tex]
[tex]\[ \text{Fraction length smooth} = \frac{\frac{ m + \mu mg - F}{\mu}}{mg}. \][/tex]
[tex]\[ \text{Fraction length smooth} = \frac{ m + \mu mg - F}{\mu mg}. \][/tex]
Simplifying:
[tex]\[ \text{Fraction length smooth} = \frac{ F }{ \mu mg}. \][/tex]
This matches Option (A). Therefore, the fraction of length on the smooth surface is:
[tex]\[ \boxed{\frac{F}{\mu m g}}. \][/tex]
