An open knife edge of mass M is dropped from a height h on a wooden floor. If the blade penetrates S into the wood, the average resistance offered by the wood to the blade is

Options

A. M g
B. M g `(1 + (h)/(s))`
C. M g `(1 - (h)/(S))`
D. M g ` (1 + (h)/(S))^(2)`

Explanation

To find the average resistance offered by the wood to the blade of the knife, we can follow these steps:

### Step 1: Determine the velocity of the knife just before impact
When the knife is dropped from a height \( h \), it falls freely under gravity. The initial velocity \( u \) is 0, and the final velocity \( v \) just before it hits the wooden floor can be calculated using the equation of motion:

\[ v^2 = u^2 + 2gh \]

Substituting \( u = 0 \):

\[ v^2 = 0 + 2gh \implies v = \sqrt{2gh} \]

### Step 2: Calculate the deceleration while penetrating the wood
When the knife penetrates the wood to a depth \( S \), it comes to a stop. We can use the same equation of motion to find the deceleration \( a \) while it is penetrating the wood. Here, the initial velocity \( u \) is \( \sqrt{2gh} \), the final velocity \( v \) is 0, and the distance \( s \) is \( -S \):

\[ 0 = (\sqrt{2gh})^2 + 2a(-S) \]

This simplifies to:

\[ 0 = 2gh - 2aS \implies 2aS = 2gh \implies a = \frac{gh}{S} \]

### Step 3: Calculate the net force acting on the knife
The forces acting on the knife while it is penetrating the wood include the weight of the knife \( mg \) acting downwards and the resistance force \( F_r \) of the wood acting upwards. According to Newton's second law, the net force can be expressed as:

\[ F_{\text{net}} = ma \]

The net force acting on the knife can be expressed as:

\[ F_r - mg = -ma \]

Rearranging gives:

\[ F_r = mg + ma \]

### Step 4: Substitute the value of \( a \)
From Step 2, we found that \( a = \frac{gh}{S} \). Substituting this into the equation for \( F_r \):

\[ F_r = mg + m\left(\frac{gh}{S}\right) \]

Factoring out \( mg \):

\[ F_r = mg\left(1 + \frac{h}{S}\right) \]

### Final Result
Thus, the average resistance offered by the wood to the blade of the knife is:

\[ F_r = mg\left(1 + \frac{h}{S}\right) \]