Problem Statement
A square plate of size 1 m x 1 m and weighing 350 N slides down an inclined plane with a uniform velocity of 1.5 m/s. The inclined plane is laid on a slope of 5 vertical to 12 horizontal and has an oil film of 1 mm thickness. Calculate the dynamic viscosity of oil.
Given Data
- Plate Area, \(A = 1 \, \text{m} \times 1 \, \text{m} = 1 \, \text{m}^2\)
- Weight of Plate, \(W = 350 \, \text{N}\)
- Uniform Velocity, \(u = 1.5 \, \text{m/s}\)
- Slope: 5 (Vertical) to 12 (Horizontal)
- Oil Film Thickness, \(dy = 1 \, \text{mm}\)
Solution
1. Determine the Angle of Inclination (\(\theta\))
The slope forms a right-angled triangle. We can find \(\sin\theta\) directly without calculating the angle itself.
2. Calculate the Driving Force (\(F_{\text{driving}}\))
This is the component of the plate’s weight acting parallel to the inclined plane.
3. Apply Force Equilibrium
Since the plate moves at a uniform velocity, the driving force is balanced by the opposing viscous drag force (\(F_{\text{drag}}\)).
4. Calculate Dynamic Viscosity (\(\mu\))
The drag force is related to viscosity by \( F_{\text{drag}} = \mu \frac{u}{dy} A \). We rearrange this to solve for \(\mu\).
Converting to Poise (1 N·s/m² = 10 Poise):
The dynamic viscosity of the oil is approximately \( \mu \approx 0.898 \, \text{Poise} \).
Explanation of the Physics
1. Force Balance on an Incline:
The core principle is that the plate is in equilibrium because it moves at a constant velocity. This means all forces acting along the incline must cancel out. The force pulling the plate down the slope (a component of its weight) is perfectly matched by the viscous friction force from the oil pulling it up the slope.
2. Viscous Drag:
As the plate slides, it shears the oil film between it and the stationary plane. The oil’s internal resistance to this motion, quantified by its dynamic viscosity (\(\mu\)), creates a drag force. This force is proportional to the viscosity, the contact area, and the velocity, and inversely proportional to the film thickness.
Physical Meaning
The calculated dynamic viscosity (\(\mu \approx 0.898\) Poise) is a measure of the oil’s “thickness” or resistance to flow. This specific value is what is required to create just enough fluid friction to support the plate’s weight component on the incline, allowing it to slide at a steady 1.5 m/s.
If the oil were thicker (higher viscosity), the drag force would be greater, and the plate would slide down more slowly or stop. If the oil were thinner (lower viscosity), the drag would be insufficient to counteract the driving force, and the plate would accelerate down the plane. This demonstrates how viscosity can be used to control motion and dissipate energy in mechanical systems.
