91Ó°ÊÓ

A \(0.8\)-lbm object traveling at \(200 \mathrm{ft} / \mathrm{s}\) enters a viscous liquid and is essentially brought to rest before it strikes the side. What is the increase in internal energy, taking the object and the liquid as the system? Neglect the potential energy change. SOLUTION Conservation of energy requires that the sum of the kinetic energy and internal energy remain constant since we are neglecting the potential energy change. This allows us to write $$ E_{1}=E_{2} \quad \frac{1}{2} m V_{1}^{2}+U_{1}=\frac{1}{2} m V_{2}^{2}+U_{2} $$ The final velocity \(V_{2}\) is zero, so that the increase in internal energy \(\left(U_{2}-U_{1}\right)\) is given by $$ U_{2}-U_{1}=\frac{1}{2} m V_{1}^{2}=\left(\frac{1}{2}\right)(0.8 \mathrm{lbm})\left(200^{2} \mathrm{ft}^{2} / \mathrm{s}^{2}\right)=16.000 \mathrm{lbm}-\mathrm{ft}^{2} / \mathrm{s}^{2} $$ We can convert the above units to ft-lbf, the usual units on energy: $$ U_{2}-U_{1}=\frac{16,000 \mathrm{lbm}-\mathrm{ft}^{2} / \mathrm{s}^{2}}{32.2 \mathrm{lbm}-\mathrm{ft} / \mathrm{s}^{2}-\mathrm{lbf}}=497 \mathrm{ft}-\mathrm{lbf} $$

Step by step solution

01

Identify the Object’s Initial and Final Conditions

The mass of the object is 0.8 lbm, the initial velocity is 200 ft/s, and the final velocity is 0 ft/s. Potential energy change is neglected.

02

Apply the Conservation of Energy Principle

Since potential energy change is neglected and the object comes to rest in the viscous liquid, conserve energy with kinetic and internal energy components: \[ E_1 = E_2 \quad \frac{1}{2} m V_1^2 + U_1 = \frac{1}{2} m V_2^2 + U_2 \]

03

Initial and Final Kinetic Energy

The initial and final kinetic energy can be calculated. Since the final velocity (\(V_2\)) is zero, the increase in internal energy (\(U_2 - U_1\)) can be given by: \[U_2 - U_1 = \frac{1}{2} m V_1^2 \]

04

Calculate the Increase in Internal Energy

Substitute the values into the equation: \[ U_2 - U_1 = \left( \frac{1}{2} \right) (0.8 \text{ lbm}) \left( 200^2 \text{ ft}^2 / \text{s}^2 \right) = 16,000 \text{ lbm-ft}^2 / \text{s}^2 \]

05

Convert Units to ft-lbf

Convert the computed internal energy from lbm-ft^2/s^2 to ft-lbf: \[ U_2 - U_1 = \frac{16,000 \text{ lbm-ft}^2 / \text{s}^2}{32.2 \text{ lbm-ft}/\text{s}^2-\text{lbf}} = 497 \text{ ft-lbf} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy

Understanding kinetic energy is pivotal in thermodynamic problems. Kinetic energy represents the energy an object possesses due to its motion. It's given by the equation: \[ KE = \frac{1}{2} m V^2 \] where \(m\) is the mass and \(V\) is the velocity of the object. In our problem, the object's initial kinetic energy can be calculated using the given values of mass and velocity. This energy converts into internal energy when the object comes to rest in the viscous liquid, as its velocity drops to zero.

Internal Energy

Internal energy is a form of energy related to the microscopic components of a system, including kinetic energy of particles and potential energy from particle interactions. When the object in our problem stops moving in the viscous liquid, its kinetic energy is converted to internal energy. This relationship comes from the Conservation of Energy principle, where the increase in internal energy \(U_2 - U_1\) is equal to the lost kinetic energy:

Unit Conversion in Thermodynamics

In thermodynamics, unit conversion is crucial for accurate calculations. In our problem, the internal energy is initially calculated in \( \text{lbm} \cdot \text{ft}^2 / \text{s}^2\). However, to make it more interpretable, we convert it to \(\text{ft-lbf}\) using the conversion factor \(32.2 \ \text{lbm} \cdot \text{ft} / \text{s}^2 - \text{lbf}\). Proper unit conversion ensures we use consistent and understandable units in our computations.

Viscous Damping

Viscous damping refers to the resistance force exerted by a fluid on a moving object. This force leads to the dissipation of kinetic energy into internal energy and heat. In our problem, the viscous liquid slows down the object and brings it to rest, effectively converting its kinetic energy into internal energy within the object-liquid system. Understanding viscous damping helps to comprehend how energy is conserved and transformed in real-life scenarios involving fluids.

Energy Principles

The conservation of energy is fundamental in thermodynamics. It states that energy cannot be created or destroyed, only transformed from one form to another. In our exercise, this principle is used to equate the initial kinetic energy of the object to the increase in internal energy after the object comes to rest. By applying this principle, we are able to determine that all the initial kinetic energy translates into an increase in internal energy, thereby illustrating energy conservation effectively.