A roller of mass `300kg` and of radius `50cm` lying on horizontal floor is resting against a step of height `20cm`. The minimum horizontal force to be applied on the roller passing through its centre to turn the roller on to the step is

Options

A. `980N`
B. `1960 N`
C. `2940 N`
D. `3920 N`

Explanation

To find the minimum horizontal force required to turn the roller onto the step, we can follow these steps:

### Step 1: Understand the Problem
We have a roller of mass \( m = 300 \, \text{kg} \) and radius \( r = 0.5 \, \text{m} \) resting against a step of height \( h = 0.2 \, \text{m} \). We need to determine the minimum horizontal force \( F \) needed to make the roller climb over the step.

### Step 2: Identify the Forces Acting on the Roller
The forces acting on the roller include:
- The weight of the roller \( W = mg \), where \( g \approx 9.81 \, \text{m/s}^2 \).
- The applied horizontal force \( F \).
- The normal reaction force at the point of contact with the step.

### Step 3: Calculate the Weight of the Roller
First, we calculate the weight of the roller:
\[ W = mg = 300 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 2943 \, \text{N} \]

### Step 4: Determine the Geometry of the Situation
When the roller begins to climb the step, the center of the roller will be lifted. The vertical distance from the center of the roller to the top of the step is:
\[ \text{Vertical distance} = r - h = 0.5 \, \text{m} - 0.2 \, \text{m} = 0.3 \, \text{m} \]

### Step 5: Set Up the Moment Equation
To find the minimum force \( F \), we can set up the moment about the edge of the step. The moment due to the weight of the roller about the edge of the step must equal the moment due to the applied force \( F \).

The distance from the edge of the step to the center of the roller is:
\[ d = \sqrt{r^2 - h^2} = \sqrt{(0.5)^2 - (0.2)^2} = \sqrt{0.25 - 0.04} = \sqrt{0.21} \approx 0.458 \, \text{m} \]

The moment due to the weight of the roller about the edge of the step is:
\[ \text{Moment due to weight} = W \cdot d = 2943 \, \text{N} \cdot 0.458 \, \text{m} \]

The moment due to the applied force \( F \) is:
\[ \text{Moment due to force} = F \cdot h = F \cdot 0.2 \, \text{m} \]

### Step 6: Set the Moments Equal
Setting the moments equal gives us:
\[ 2943 \cdot 0.458 = F \cdot 0.2 \]

### Step 7: Solve for the Force \( F \)
Now, we can solve for \( F \):
\[ F = \frac{2943 \cdot 0.458}{0.2} \approx \frac{1341.834}{0.2} \approx 6709.17 \, \text{N} \]

### Final Answer
The minimum horizontal force required to turn the roller onto the step is approximately \( 6709.17 \, \text{N} \).

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