Problem 20 - Thomas Calculus in SI Units

Question

A right-circular cylindrical tank of height 3 m and radius 1.5 m is lying horizontally and is full of diesel fuel weighing \(8300 \mathrm{N} / \mathrm{m}^{3}\). How much work is required to pump all of the fuel to a point \(4.5 \mathrm{m}\) above the top of the tank?

Step-by-Step Solution

Answer

The work required is approximately 785378.55 J.

Step 1: Formula for Work in Mechanics

The work required to lift the fuel involves calculating the force (weight of the fuel) multiplied by the distance it must be lifted. The basic formula for work is given by \( W = F \cdot d \), where \( F \) is the force and \( d \) is the distance.

Step 2: Volume of the Fuel

First, calculate the volume of the diesel fuel in the cylindrical tank. The volume \( V \) of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Given: \( r = 1.5 \) m and \( h = 3 \) m, thus:\[ V = \pi (1.5)^2 (3) = 6.75\pi \text{ m}^3 \]

Step 3: Weight of the Diesel Fuel

The weight of the diesel fuel can be calculated by multiplying its volume by the weight per unit volume. The total weight \( W_t \) is:\[ W_t = \text{Volume} \times \text{Weight per unit volume} = 6.75\pi \times 8300 \text{ N/m}^3 \]

Step 4: Simplified Work Calculation with Weight and Distance

The distance the fuel needs to be lifted is 4.5 m (to a point above the tank). Therefore, the work \( W \) required to pump the fuel is:\[ W = W_t \times 4.5 = (6.75\pi \times 8300) \times 4.5 \]

Step 5: Calculate the Final Work Done

Substitute and simplify the expression to find the total work:\[ W = 6.75 \times 3.1416 \times 8300 \times 4.5 \approx 785378.55 \text{ J} \]

Key Concepts

Understanding a Cylindrical Tank

A cylindrical tank is a 3D shape with straight parallel sides and circular ends. In this exercise, the tank is a right circular cylinder lying horizontally. The dimensions of the tank are important in determining the volume and further calculations. The radius of the base is 1.5 meters, and the height (or length in the horizontal position) is 3 meters. Cylindrical tanks are commonly used for storage, including holding fuels like diesel. Knowing the tank's orientation helps us understand how the liquid inside is distributed. Frictionless, horizontal orientation allows easy computation of volume and weight.

Calculating Volume of a Cylinder

To find how much space is inside a cylindrical tank, you need to calculate the volume. The formula for the volume of a cylinder is: \( V = \pi r^2 h \), where:

• \( r \) is the radius of the circular base.
• \( h \) is the height when standing; length when lying horizontally.

In our case, the radius \( r \) is 1.5 meters, and the height \( h \) is 3 meters. By plugging these values into the formula, you get:\[V = \pi (1.5)^2 \times 3 = 6.75\pi \, \text{cubic meters}\]Volume is crucial as it directly affects how much fuel (or any material inside) is contained. More volume means more weight to lift, hence more work is needed.

Determining the Weight of the Fuel

Once you have the volume of the fuel, the next step is to find its weight. Fuel weight is calculated by multiplying its volume by the weight per unit volume. For diesel, this weight density is approximately 8300 N/m³.So, the total weight \( W_t \) can be derived as:\[W_t = 6.75\pi \times 8300 \, \text{Newtons}\]Knowing the weight is pivotal because, in mechanics, it represents the force needed to lift the fuel. Essentially, a heavier weight requires more work when being moved against gravity.

Considering Lifting Distance for Work Calculation

The work required to move an object is calculated using the formula: \( W = F \times d \), where \( F \) is force, and \( d \) is the distance moved in the direction of the force. In this case, the force is the weight of the diesel fuel. The lifting distance is 4.5 meters, which is the height above the tank's top level where the fuel is pumped. Hence, the work done to lift the fuel is:\[W = (6.75\pi \times 8300) \times 4.5\]This step considers how far the substance needs to be moved, determining the total effort. Distance affects work linearly. The greater the distance, the more effort required to complete the task.