An oil film of thickness 1.5 mm is used for lubrication between a square plate of size 0.9 m × 0.9 m and an inclined plane having an angle of inclination 20°. The weight of the square plate is 392.4 N and it slides down the plane with a uniform velocity of 0.2 m/s. Find the dynamic viscosity of the oil.

Dynamic Viscosity on an Inclined Plane

Problem Statement

Given Data

Oil Film Thickness, $dy = 1.5 \, \text{mm} = 1.5 \times 10^{-3} \, \text{m}$
Plate Size: 0.9 m × 0.9 m
Angle of Inclination, $\theta = 20^\circ$
Weight of Plate, $W = 392.4 \, \text{N}$
Uniform Velocity, $u = 0.2 \, \text{m/s}$

Solution

1. Calculate the Area of the Plate

The surface area of the square plate in contact with the oil is:

$$ A = 0.9 \, \text{m} \times 0.9 \, \text{m} $$ $$ A = 0.81 \, \text{m}^2 $$

2. Determine the Driving Force

The force causing the plate to slide down the incline is the component of its weight acting parallel to the plane.

$$ F_{\text{driving}} = W \sin\theta $$ $$ F_{\text{driving}} = 392.4 \, \text{N} \times \sin(20^\circ) $$ $$ F_{\text{driving}} \approx 134.20 \, \text{N} $$

3. Apply Force Equilibrium

Since the plate moves at a uniform velocity, the system is in equilibrium. The driving force must be balanced by the opposing viscous drag force from the oil film.

$$ F_{\text{drag}} = F_{\text{driving}} $$ $$ F_{\text{drag}} \approx 134.20 \, \text{N} $$

4. Relate Drag Force to Viscosity

The drag force is the product of shear stress ($\tau$) and area ($A$). Using Newton's law of viscosity, we can express this in terms of the unknown viscosity $\mu$.

$$ F_{\text{drag}} = \tau \times A = \left(\mu \frac{du}{dy}\right) \times A $$

Now, we can set the forces equal and solve for $\mu$:

$$ W \sin\theta = \mu \frac{u}{dy} A $$ $$ \mu = \frac{W \sin\theta \times dy}{u \times A} $$ $$ \mu = \frac{134.20 \, \text{N} \times 1.5 \times 10^{-3} \, \text{m}}{0.2 \, \text{m/s} \times 0.81 \, \text{m}^2} $$ $$ \mu \approx 1.2426 \, \text{N s/m}^2 $$

5. Convert Viscosity to Poise

To get the final answer in the requested units (1 N·s/m² = 10 poise):

$$ \mu_{\text{poise}} = \mu_{\text{SI}} \times 10 $$ $$ \mu = 1.2426 \, \text{N s/m}^2 \times 10 $$ $$ \mu \approx 12.43 \, \text{poise} $$

Final Result:

The dynamic viscosity of the oil is $ \mu \approx 12.43 \, \text{poise} $.

Explanation of the Physics

1. Force Balance:
The key to this problem is understanding the balance of forces. The plate slides at a uniform velocity, which means its acceleration is zero. According to Newton's First Law, this implies that the net force on the plate is zero. The gravitational force pulling the plate down the incline is perfectly counteracted by the frictional (viscous) force from the oil film pulling it up the incline.

2. Gravitational Component:
The weight of the plate ($W$) acts vertically downwards. On an inclined plane, this weight is resolved into two components: one perpendicular to the plane ($W\cos\theta$) and one parallel to it ($W\sin\theta$). Only the parallel component contributes to the motion down the plane.

3. Viscous Force:
The oil between the plate and the plane is sheared as the plate moves. The oil's internal resistance to this shearing motion creates a drag force. This force is dependent on the oil's viscosity ($\mu$), the velocity of the plate ($u$), the thickness of the oil film ($dy$), and the contact area ($A$).

Physical Meaning

The calculated dynamic viscosity ($\mu$) is a fundamental property of the oil that quantifies its "thickness" or resistance to flow. In this scenario, the viscosity is precisely what's needed to generate enough fluid friction to prevent the plate from accelerating down the slope.

If a less viscous (thinner) oil were used, the drag force would be lower, and the plate would accelerate down the plane. Conversely, if a more viscous (thicker) oil were used, the drag force would be greater than the gravitational component, and the plate would either slow down or require an additional external force to maintain the same speed of 0.2 m/s. This principle is critical in designing systems where lubrication is used to control motion and dissipate energy, such as in shock absorbers or hydraulic dashpots.