A man holds a 2 kg book between his palms. so that each hand exerts the same horizontal force on the book. The coefficient of static friction between the palms and the book is 0.4 and g = `10ms^(-2)`. If the book is prevented from falling, the least force exerted by each hand on the book is

Options

A. 50 N
B. 25 N
C. 75 N
D. 100 N

Answer

To solve the problem step by step, we will analyze the forces acting on the book and apply the concepts of static friction and equilibrium.

### Step 1: Identify the forces acting on the book
The forces acting on the book are:
- The weight of the book (downward)
- The normal forces exerted by each hand (upward)
- The static frictional forces exerted by each hand (upward)

### Step 2: Calculate the weight of the book
The weight (W) of the book can be calculated using the formula:
\[ W = mg \]
where:
- \( m = 2 \, \text{kg} \) (mass of the book)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating the weight:
\[ W = 2 \, \text{kg} \times 10 \, \text{m/s}^2 = 20 \, \text{N} \]

### Step 3: Set up the equilibrium condition
Since the book is held in equilibrium (not falling), the total upward forces must equal the downward force (weight of the book). The upward forces consist of the normal forces and the frictional forces.

Let \( N \) be the normal force exerted by each hand. The total upward force due to the normal forces from both hands is \( 2N \).

The static frictional force \( f_s \) can be expressed as:
\[ f_s = \mu N \]
where \( \mu = 0.4 \) (coefficient of static friction).

### Step 4: Write the equilibrium equation
The total upward force can be expressed as:
\[ 2N + 2f_s = W \]

Substituting \( f_s \):
\[ 2N + 2(\mu N) = W \]
\[ 2N + 2(0.4N) = 20 \, \text{N} \]
\[ 2N + 0.8N = 20 \, \text{N} \]
\[ 2.8N = 20 \, \text{N} \]

### Step 5: Solve for the normal force \( N \)
Now, we can solve for \( N \):
\[ N = \frac{20 \, \text{N}}{2.8} \]
\[ N = \frac{200}{28} \]
\[ N = 7.14 \, \text{N} \]

### Step 6: Calculate the least force exerted by each hand
The least force exerted by each hand on the book is equal to the normal force \( N \):
\[ \text{Force by each hand} = N = 7.14 \, \text{N} \]

### Final Answer
Thus, the least force exerted by each hand on the book is approximately **7.14 N**.