Problem 69 - College Physics

Question

An open cylindrical tank of acid rests at the edge of a table 1.4 \(\mathrm{m}\) above the floor of the chemistry lab. If this tank springs a small hole in the side at its base, how far from the foot of the table will the acid hit the floor if the acid in the tank is 75 \(\mathrm{cm}\) deep?

Step-by-Step Solution

Answer

The acid will hit the floor approximately 2.03 meters from the foot of the table.

Step 1: Understand the Problem

We have a cylindrical tank containing acid at the edge of a table. The tank height from the base to its top is 1.4 m and the acid level is 0.75 m inside the tank. A small hole in the side of the tank at its base lets the acid escape. We are to find the horizontal distance from the table to where the acid hits the floor.

Step 2: Calculate the Velocity of Fluid Leaving the Tank

To find the velocity of the acid as it exits the tank, use Torricelli's law: \( v = \sqrt{2gh} \), where \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity) and \( h = 0.75 \, \text{m} \) (depth of acid). Thus, \( v = \sqrt{2 \times 9.81 \times 0.75} \approx 3.84 \, \text{m/s} \).

Step 3: Determine the Time for the Acid to Hit the Ground

Use the formula for free fall: \( t = \sqrt{\frac{2d}{g}} \), where \( d = 1.4 \, \text{m} \) is the distance from the base of the tank to the floor. Substitute \( g = 9.81 \, \text{m/s}^2 \): \( t = \sqrt{\frac{2 \times 1.4}{9.81}} \approx 0.53 \, \text{s} \).

Step 4: Calculate the Horizontal Distance

The horizontal distance \( x \) is given by \( x = vt \), where \( v = 3.84 \, \text{m/s} \) (velocity of fluid) and \( t = 0.53 \, \text{s} \) (time calculated). Thus, \( x = 3.84 \times 0.53 \approx 2.03 \, \text{m} \).

Step 5: Conclusion

The acid will hit the floor approximately 2.03 meters from the foot of the table.

Key Concepts

Torricelli's Law

Torricelli's Law is a fundamental principle of fluid dynamics that helps us understand the speed at which a fluid exits a hole in a container. This principle is based on the conservation of energy. It shows how potential energy is converted into kinetic energy as fluid flows under gravity. Torricelli's equation is expressed as \( v = \sqrt{2gh} \), where:

• \( v \) represents the velocity of the fluid exiting the hole.
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \, \text{m/s}^2 \) on Earth.
• \( h \) is the height of the liquid column above the hole.

In our exercise, we used \( h = 0.75 \, \text{m} \) for the depth of acid to determine its velocity as it leaves the tank. This simple calculation allows predictions about how quickly fluids will move, which is vital in designing spillways, pipes, and fluid containers.

Cylindrical Tank

A cylindrical tank is a common structure for storing liquids. It has a circular base and straight sides, making it practical for holding substances like water, oil, or, in our exercise, acid. When considering a hole at the base, understanding the shape is crucial for fluid flow calculations. In this scenario:

• The shape of the tank influences the distribution and pressure of the liquid inside.
• The container's vertical sides mean that any hole at the base releases fluid directly related to the height of liquid above it — key for using Torricelli's Law.

Cylindrical tanks offer straightforward models for studying fluid dynamics because their geometric simplicity allows for easy application of mathematical principles relating to pressure and flow.

Free Fall Equations

Free fall equations describe the motion of objects under the influence of gravity alone. In our exercise, these equations help determine how long the acid takes to hit the floor once it exits the tank. The key free fall equation utilized here is:

• \( t = \sqrt{\frac{2d}{g}} \)

Where:

• \( t \) is the time it takes for the object to fall to the ground.
• \( d \) is the vertical drop distance from the hole to the ground (1.4 m here).
• \( g \) is the standard gravitational acceleration (9.81 m/s²).

This simple equation helps predict and understand the motion of free-falling objects, allowing the calculation of both time and final velocity when gravity is the sole acting force.

Fluid Dynamics

Fluid dynamics is the study of fluids and their behaviors, either at rest or in motion. It provides insight into how forces affect fluids, enabling us to predict outcomes in cases like the tank exercise. Key concepts from fluid dynamics include:

• Understanding flow rate and pressure differences, crucial for predicting how fluid behaves when released from a container.
• Recognizing the impact of container shape and size on fluid behavior — a cylindrical shape, for example, provides consistent pressure distribution along the vertical axis.
• Applying principles like Torricelli's Law to determine velocity and subsequent movement patterns.

This field combines physics and mathematics to resolve practical problems involving fluids, ensuring efficient design and analysis of systems like water supply networks and chemical tanks.