Calculating Rolling Resistance Force

This question involves calculating the force required to overcome rolling resistance when a steel wheel rolls on a horizontal steel rail. Rolling resistance is a force that resists the motion when a body (like a wheel) rolls on a surface. It's caused by factors like deformation of the wheel and the surface.

Understanding the Given Parameters

We are provided with the following information:

• Wheel Diameter ($D$): 600 mm
• Load Carried ($W$): 500 N
• Coefficient of Rolling Resistance ($a$): 0.3 mm

Determining the Wheel Radius

The radius ($R$) of the wheel is half of its diameter ($D$).

The formula is: $R = \frac{D}{2}$

Substituting the given diameter:

$R = \frac{600 \, \text{mm}}{2}$

$R = 300 \, \text{mm}$

Applying the Rolling Resistance Formula

The force ($F$) required to overcome rolling resistance is calculated using the coefficient of rolling resistance ($a$), the load ($W$), and the wheel radius ($R$). The formula used when the coefficient is given in units of length (like mm) is:

$$ F = W \times \frac{a}{R} $$

Where:

• $F$ is the force required (in N)
• $W$ is the load (in N)
• $a$ is the coefficient of rolling resistance (in mm)
• $R$ is the radius of the wheel (in mm)

It's important that the units for '$a$' and '$R$' are consistent (both in mm in this case).

Step-by-Step Force Calculation

Now, let's substitute the known values into the formula:

• $W = 500 \, \text{N}$
• $a = 0.3 \, \text{mm}$
• $R = 300 \, \text{mm}$

$$ F = 500 \, \text{N} \times \frac{0.3 \, \text{mm}}{300 \, \text{mm}} $$

First, calculate the ratio $\frac{a}{R}$:

$$ \frac{0.3}{300} = 0.001 $$

Now, multiply this ratio by the load ($W$):

$$ F = 500 \, \text{N} \times 0.001 $$

$$ F = 0.5 \, \text{N} $$

Final Result

The calculated force necessary to roll the wheel along the rail is 0.5 N. This matches the first option provided.