A block weighing 10 lbf and having dimensions 25 cm on each edge is pulled up an inclined surface on which there is a film of SAE 10W oil at 100F. If the speed of the block is 1 m/s and the oil film is 50 um thick, find the force required to pull the block. The surface is inclined at an angle of 25 deg from the horizontal
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PK Bot’s answer:
Assume that the viscous force arises from a simple Couette‐flow shear stress in the thin oil film and that the block’s bottom face (area A) is in full contact with the oil. In Couette flow the shear stress is
\tau=\mu\frac{U}{h},
where
• U=1~m/s is the speed of the block relative to the stationary surface,
• h=50\times10^{-6}~m is the oil film thickness, and
• \mu is the dynamic viscosity of SAE 10W oil at 100°F.
In the absence of a provided oil viscosity, we assume a representative value of
\mu=0.20~Pa\cdot s,
which is a reasonable approximation at 100°F.
The block has dimensions of 0.25 m on a side so its bottom area is
A=0.25^2=0.0625~m^2.
Then the viscous force is
F_{\text{viscous}}=\tau A=\mu\frac{U}{h}A
=\;0.20\frac{1}{50\times10^{-6}}(0.0625)
=\;0.20\,(20\,000)(0.0625)
=\;250~N.
In addition, the block’s weight provides a component down the incline. A 10 lbf weight is approximately
W\approx44.5~N.
On an incline of 25° the gravitational component along the slope is
F_g=W\sin{25^\circ}\approx44.5\times0.4226\approx18.8~N.
Thus the total force required to pull the block up the incline is the sum of the viscous force and the downhill gravitational component:
F_{\text{total}}\approx250+18.8\approx269~N.
Assumptions made:
• The block–surface interface is completely covered by a uniform oil film of thickness 50 µm.
• The Couette shear stress applies (i.e. the velocity gradient is linear across the film).
• \mu=0.20~Pa\cdot s for SAE 10W oil at 100°F is taken as representative.
The required force is therefore approximately 270 N.
The ansatz that the Couette shear stress applies can be partially checked by checking the Reynolds number:
\mathrm{Re}=\frac{\rho U h}{\mu}
which results in a \mathrm{Re} below 1, which confirms that the flow is laminar.
To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked their significance below. If they are marked as relevant, I encourage you to take a look at them:
Textbook sections:
- Chapter 9 Fundamentals: The force on an object is calculated by integrating the surface stress along… (RELEVANT)
- Chapter 4.2: Review of Hydrostatics (RELEVANT)
- Chapter 3.2: Hydrostatic Equation (MARGINALLY RELEVANT)
- Chapter 7.3: Integral Conservation of Momentum Part 2 (MARGINALLY RELEVANT)
- Appendix F.3: Volumetric Viscous Force Part 1 (MARGINALLY RELEVANT)
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