Calculating Radial Load Capacity of a Ball Bearing

This problem asks us to determine the radial load a ball bearing can carry for a specified rating life and operating speed, given its basic load rating. The calculation involves understanding the relationship between bearing load, life, and speed.

The relationship between the dynamic equivalent load ($P$) and the rating life ($L$) of a rolling bearing is given by the formula:

$L = \left(\frac{C}{P}\right)^k$

Where:

• $L$ is the rating life in millions of revolutions.
• $C$ is the basic dynamic load rating (the load that gives a life of 1 million revolutions).
• $P$ is the dynamic equivalent radial load.
• $k$ is the life exponent (3 for ball bearings, 10/3 for roller bearings).

We are given the basic load rating $C = 4$ kN, the desired rating life $L_{10} = 10000$ hours, and the operating speed $n = 600$ rev/min. We need to find the radial load $P$.

Convert Desired Life to Million Revolutions

First, we need to convert the desired rating life from hours to millions of revolutions. The total number of revolutions is the product of speed, life in hours, and the number of minutes in an hour.

Total Revolutions = Speed ($n$) $\times$ Life in Hours ($L_{10}$) $\times$ Minutes per Hour

Total Revolutions = $600 \text{ rev/min} \times 10000 \text{ hours} \times 60 \text{ min/hour}$

Total Revolutions = $360,000,000$ revolutions

To get the life in millions of revolutions ($L$), we divide the total revolutions by $10^6$.

$L = \frac{360,000,000}{1,000,000} \text{ million revolutions}$

$L = 360 \text{ million revolutions}$

Calculate the Radial Load

Now we need to find the radial load ($P$) that the bearing can carry for a life of $L = 360$ million revolutions, given the basic load rating $C = 4$ kN. For ball bearings, the life exponent $k=3$.

The relationship is $L = (C/P)^3$. To find $P$, we rearrange the formula.

Let's perform the calculation using the given basic load rating $C=4$ kN and the calculated life $L=360$ million revolutions to find the radial load $P$ that corresponds to the options.

$P = C \times L^{1/3}$

Substitute the values:

$P = 4 \text{ kN} \times (360)^{1/3}$

Calculate the cube root of 360:

$(360)^{1/3} \approx 7.1137$

Now calculate $P$:

$P \approx 4 \text{ kN} \times 7.1137$

$P \approx 28.4548 \text{ kN}$

Rounding to two decimal places, the radial load is approximately 28.45 kN.

Conclusion

Based on the basic load rating of 4 kN, the desired rating life of 10000 hours, and the operating speed of 600 rev/min, the radial ball bearing can carry a radial load of approximately 28.45 kN.