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

Options

A. `mg (1 + (h)/(d))`
B. `mg (1 + (h)/(d))^(2)`
C. `mg(1-h/d)`
D. `mg (1 +d /(h))`

Explanation

To solve the problem of an open knife of mass \( m \) being dropped from a height \( h \) and penetrating a wooden floor to a depth \( d \), we can follow these steps:

### Step 1: Understand the Energy Conversion
When the knife is dropped, it has gravitational potential energy due to its height \( h \). As it falls, this potential energy is converted into work done against the resistance offered by the wood when the knife penetrates to a depth \( d \).

### Step 2: Calculate the Total Distance Fallen
The total distance the knife falls before it comes to rest after penetrating the wood is the sum of the height \( h \) from which it was dropped and the depth \( d \) into which it penetrates:
\[ \text{Total distance} = h + d \]

### Step 3: Calculate the Potential Energy
The potential energy (\( PE \)) of the knife at the height \( h \) is given by:
\[ PE = mgh \]
However, since the knife penetrates into the wood, we need to consider the total distance fallen:
\[ PE = mg(h + d) \]

### Step 4: Work Done Against Resistance
The work done (\( W \)) by the knife on the wooden block as it penetrates is equal to the average resistance force (\( F \)) times the distance penetrated (\( d \)):
\[ W = F \cdot d \]

### Step 5: Set Potential Energy Equal to Work Done
Since the potential energy is converted into work done against the resistance, we can set the two equations equal to each other:
\[ mg(h + d) = F \cdot d \]

### Step 6: Solve for Average Resistance Force
Rearranging the equation to solve for the average resistance force \( F \):
\[ F = \frac{mg(h + d)}{d} \]

### Step 7: Simplify the Expression
We can simplify the expression for \( F \):
\[ F = mg \left( \frac{h}{d} + 1 \right) \]

### Final Answer
Thus, the average resistance offered by the wood to the knife edge is:
\[ F = mg \left( 1 + \frac{h}{d} \right) \]