The free surface of oil in a tanker, at rest, is horizontal. If the tanker starts accelerating, the free surface will be tilted by an angle theta. If the acceleration is a m/s², wh... The free surface of oil in a tanker, at rest, is horizontal. If the tanker starts accelerating, the free surface will be tilted by an angle theta. If the acceleration is a m/s², what will be the slope of the free surface?
Understand the Problem
The question is asking to determine the angle of tilt (theta) of the free surface of oil in a tanker that accelerates at a given acceleration (a). To solve this, we can apply the principles of fluid mechanics that describe how the free surface of a liquid adjusts to maintain equilibrium under acceleration.
Answer
$$ \theta = \tan^{-1}\left(\frac{a}{g}\right) $$
Answer for screen readers
$$ \theta = \tan^{-1}\left(\frac{a}{g}\right) $$
Steps to Solve
- Identify the forces acting on the oil
When the tanker accelerates, two main forces affect the oil:
- The gravitational force acting downward ($W = mg$)
- The inertial force acting horizontally due to the tanker’s acceleration ($F_{inertia} = ma$)
- Set up the free body diagram
If the tanker accelerates to the right, the oil surface will tilt. The angle of tilt ($\theta$) can be represented in terms of the forces acting on it.
- Use trigonometry to relate forces
The following relation holds based on the balance of forces: $$ \tan(\theta) = \frac{F_{inertia}}{W} = \frac{ma}{mg} $$ This simplifies to: $$ \tan(\theta) = \frac{a}{g} $$
- Calculate theta
To find the angle $\theta$, use the arctangent function: $$ \theta = \tan^{-1}\left(\frac{a}{g}\right) $$
$$ \theta = \tan^{-1}\left(\frac{a}{g}\right) $$
More Information
The angle of tilt, $\theta$, is determined by the ratio of the acceleration of the tanker ($a$) to the gravitational acceleration ($g$, approximately $9.81 , \text{m/s}^2$). This relationship illustrates how the fluid surface adjusts itself to maintain equilibrium in accelerating frames.
Tips
- Forgetting to use the correct gravitational constant value ($g = 9.81 , \text{m/s}^2$).
- Mixing up acceleration and gravitational components could lead to incorrect angles.
- Not using the arctangent function correctly can lead to an incorrect understanding of the angle.
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