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 + /(h))`

Explanation

To solve the problem of a knife dropped from a height \( h \) that penetrates a wooden floor to a depth \( d \), we can use the principles of energy conservation and work done against resistance. Here’s the step-by-step solution:

### Step 1: Calculate the Potential Energy of the Knife
When the knife is dropped from a height \( h \), it has gravitational potential energy given by:
\[ PE = mgh \]
where \( m \) is the mass of the knife and \( g \) is the acceleration due to gravity.

**Hint:** Remember that potential energy is related to height and mass.

### Step 2: Determine the Total Distance Fallen
When the knife penetrates the wood to a depth \( d \), the total distance fallen is \( h + d \). Thus, the potential energy when it reaches the depth \( d \) is:
\[ PE_{total} = mg(h + d) \]

**Hint:** Consider the total distance the knife travels before it comes to rest.

### Step 3: Work Done Against the Resistance
The work done against the resistance offered by the wood as the knife penetrates to depth \( d \) can be expressed as:
\[ W = F \cdot d \]
where \( F \) is the average resistance force offered by the wood.

**Hint:** Work done is the product of force and distance moved in the direction of the force.

### Step 4: Apply the Work-Energy Principle
According to the work-energy principle, the work done against the resistance is equal to the loss of potential energy:
\[ F \cdot d = mg(h + d) \]

**Hint:** Set the work done equal to the change in energy to find the force.

### Step 5: Solve for the Average Resistance Force \( F \)
Rearranging the equation gives:
\[ F = \frac{mg(h + d)}{d} \]
This can be simplified to:
\[ F = mg \left(1 + \frac{h}{d}\right) \]

**Hint:** Factor out common terms to simplify the expression.

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

### Final Answer
The average resistance offered by the wood to the knife edge is \( mg \left(1 + \frac{h}{d}\right) \).

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