Work-Energy Theorem: Analyzing Particle Speed Changes

Hello! Welcome to your next lesson in Dynamics.

In our last lesson, we explored Newton's Second Law, $\sum \vec{F}=m\vec{a}$. This powerful tool allows us to find the forces on an object given its acceleration, or vice-versa, at a specific instant in time. However, if we want to find the change in an object's speed over a certain distance, we would need to integrate acceleration with respect to position, which can be cumbersome.

Today, we will learn a new and often more direct approach derived from Newton's Second Law: the Principle of Work and Energy. This method provides a powerful shortcut for solving problems that involve forces, displacement, and velocity. It's particularly useful when you don't need to know about the time taken.

This principle is a cornerstone of mechanics and has wide-ranging applications, from calculating the speed of a vehicle after its engine applies a force over a certain distance, to determining the launch velocity needed for a satellite.

By the end of this lesson, you will be able to:

• Apply the work-energy principle to analyze changes in particle speed due to applied forces.

1. The Building Blocks: Work and Kinetic Energy

Before we state the main principle, let's define its two key components: work and kinetic energy. In physics and engineering, these terms have very precise meanings.

What is Work?

Work is the energy transferred to or from an object by the application of force along a displacement.

• Work of a Constant Force: If a constant force $\vec{F}$ acts on an object that undergoes a displacement $\vec{d}$, the work done ($U$) is the product of the magnitude of the displacement and the component of the force parallel to the displacement.
  $U=(F\cos \theta )d$
  where $\theta$ is the angle between the force and displacement vectors. Work is a scalar quantity, measured in joules (J), where $1\text{J}=1\text{N}\cdot \text{m}$.

• Positive and Negative Work: Work is positive if the force component is in the same direction as the displacement (it adds energy to the object). It's negative if the force component is in the opposite direction (it removes energy). A force perpendicular to displacement does zero work. For example, friction does negative work, while a normal force on an object sliding horizontally does zero work.

• Work of Common Forces: We frequently encounter work done by weight and springs:

  • Work of Weight (Gravity): The work done by gravity depends only on the vertical change in height, $\Delta y$.
    $U_{gravity}=-W\Delta y=-mg(y_2-y_1)$
    Work is positive when the object moves down ($\Delta y<0$) and negative when it moves up ($\Delta y>0$).
  • Work of a Spring: The force from a spring varies with its stretch or compression. The work done by the spring as it goes from a stretched/compressed position $s_1$ to $s_2$ is:
    $U_{spring}=-\frac{1}{2}k(s_2^2-s_1^2)$
    Here, $k$ is the spring constant and $s$ is the position relative to the spring's unstretched length. Note the negative sign: if you stretch a spring from $s_1=0$ to $s_2=0.1$m, the spring pulls back, so it does negative work on the object it's attached to.

• Work of a Variable Force: If a force's magnitude changes with position, we must use integration to find the work done:
  $U={\int }_{s_1}^{s_2}F(s)\cos \theta ds$

The following video provides an excellent summary of how to calculate work for these different scenarios.

What is Kinetic Energy?

Kinetic energy is the energy an object possesses due to its motion. It is also a scalar quantity, measured in joules. For a particle of mass $m$ moving at a speed $v$, the kinetic energy ($T$) is:
$T=\frac{1}{2}m{v}^2$
Since mass $m$ and speed squared $v^2$ are always non-negative, kinetic energy is always a non-negative value.

2. The Principle of Work and Energy

The principle of work and energy states that the total work done by all forces acting on a particle as it moves between two positions is equal to the change in its kinetic energy.

We can write this relationship as:
$T_1+\sum U_{1-2}=T_2$

Where:

• $T_1=\frac{1}{2}m{v}_1^2$ is the initial kinetic energy of the particle.
• $T_2=\frac{1}{2}m{v}_2^2$ is the final kinetic energy of the particle.
• $\sum U_{1-2}$ is the net work done by all forces (applied forces, friction, weight, springs) as the particle moves from position 1 to position 2.

The main advantage of this principle is that it's a scalar equation that directly relates the initial and final states of motion. We don't need to determine the acceleration and integrate over time, as we would with $\sum \vec{F}=m\vec{a}$.

3. Solving Problems with the Work-Energy Principle

To apply this principle effectively, we can follow a systematic procedure.

Here's a summary of the steps:

1. Define States: Clearly identify the initial state (position 1) and final state (position 2) of the particle.
2. Draw a Free-Body Diagram: Draw the FBD to identify all forces acting on the particle. This is crucial for identifying which forces will do work.
3. Calculate Work: For each force, calculate the work it does as the particle moves from state 1 to state 2. Be careful with signs (positive, negative, or zero). Sum them to find the net work, $\sum U_{1-2}$.
4. Calculate Kinetic Energies: Write expressions for the initial ($T_1$) and final ($T_2$) kinetic energies.
5. Apply the Principle: Substitute the work and kinetic energy terms into the equation $T_1+\sum U_{1-2}=T_2$ and solve for the unknown quantity (often a final speed or a distance).

Let's watch a worked example that puts this process into practice.

Test your understanding!

A 40 kg box is initially at rest on a rough horizontal floor. A constant horizontal force of 130 N is applied to it. The coefficient of kinetic friction between the box and floor is ${\mu }_k=0.25$. What is the speed of the box after it has been pushed 5.0 m? (Use $g=9.81{\text{m/s}}^2$).

Show answer

1. States:

   • State 1: Box is at rest, $s_1=0$, $v_1=0$.
   • State 2: Box has moved, $s_2=5.0$ m, $v_2=?$.

2. FBD & Forces:

   • Applied force $P=130$ N (horizontal).
   • Weight $W=mg=40\times 9.81=392.4$ N (down).
   • Normal force $N$ (up).
   • Friction force $f_k$ (horizontal, opposing motion).

3. Calculate Work ($\sum U_{1-2}$):

   • The normal force and weight are perpendicular to the motion, so they do zero work.
   • Work done by the applied force (positive): $U_P=P\times d=130\text{N}\times 5.0\text{m}=650\text{J}$.
   • To find the work of friction, we first need $f_k$. From vertical equilibrium ($\sum F_y=0$), we have $N-W=0⟹N=392.4$ N.
   • So, the friction force is $f_k={\mu }_{k}N=0.25\times 392.4\text{N}=98.1$ N.
   • Work done by friction (negative): $U_{f_k}=-f_k\times d=-98.1\text{N}\times 5.0\text{m}=-490.5\text{J}$.
   • Net work: $\sum U_{1-2}=U_P+U_{f_k}=650-490.5=159.5\text{J}$.

4. Calculate Kinetic Energies:

   • Initial KE: $T_1=\frac{1}{2}m{v}_1^2=\frac{1}{2}(40)(0{)}^2=0$.
   • Final KE: $T_2=\frac{1}{2}m{v}_2^2=\frac{1}{2}(40)v_2^2=20v_2^2$.

5. Apply the Principle:
   $T_1+\sum U_{1-2}=T_2$
   $0+159.5=20v_2^2$
   $v_2^2=\frac{159.5}{20}=7.975(\text{m/s}{)}^2$
   $v_2=\sqrt{7.975}\approx 2.82\text{m/s}$

Answer: The final speed of the box is 2.82 m/s.

4. A Note on Potential Energy

You may have seen energy problems framed in terms of "potential energy". This is not a new principle, but rather a useful way of reorganizing the work-energy equation.

We can classify forces as either conservative or non-conservative.

• Conservative Forces: The work done by these forces does not depend on the path taken, only the start and end points. Gravity and ideal spring forces are conservative. For these forces, we can define a Potential Energy (PE or V) such that $U_{cons}=-\Delta V$.
  • Gravitational Potential Energy: $V_g=mgy$
  • Elastic Potential Energy: $V_e=\frac{1}{2}k{s}^2$
• Non-Conservative Forces: The work depends on the path. Friction and applied external forces are typical examples. We calculate their work, $U_{nc}$, directly.

By substituting $U_{cons}=V_1-V_2$ into our main equation, we get:
$T_1+(V_1-V_2)+U_{nc}=T_2$
Rearranging this gives the widely used Conservation of Energy form:
$T_1+V_1+U_{nc}=T_2+V_2$
This equation states that the initial mechanical energy ($T_1+V_1$) plus the work done by non-conservative forces equals the final mechanical energy ($T_2+V_2$). If there are no non-conservative forces ($U_{nc}=0$), then total mechanical energy is conserved: $T_1+V_1=T_2+V_2$.

5. Application to Aerospace

Let's consider an example relevant to your goals in aerospace.

This example shows the power of the work-energy method. Calculating the changing acceleration and integrating it to find the final velocity would be much more complex than simply finding the area under the force-displacement graph to get the work done.

Conclusion

In this lesson, we introduced the Principle of Work and Energy as a powerful alternative to the direct application of Newton's Second Law. It simplifies problems where we need to relate force, displacement, and velocity.

Key Takeaways:

• Work is the energy transferred by a force acting over a displacement ($U$).
• Kinetic Energy is the energy of motion ($T=\frac{1}{2}m{v}^2$).
• The Principle of Work and Energy states that the change in a particle's kinetic energy is equal to the net work done on it: $T_1+\sum U_{1-2}=T_2$.
• This is a scalar equation that provides a direct link between the initial and final states of motion, bypassing the need to calculate acceleration as an intermediate step.
• The principle can be rearranged using the concept of potential energy for conservative forces, leading to the general conservation of energy equation: $T_1+V_1+U_{nc}=T_2+V_2$.

Next Lesson Preview:

So far, we have related forces to acceleration ($F=ma$) and to displacement and velocity (Work-Energy). There is one more fundamental formulation we will explore. In the next lesson, we will introduce the Principle of Impulse and Momentum, which relates force to the time over which it acts. This is especially useful for analyzing collisions, impacts, and rocket propulsion, all of which are critical topics in aerospace engineering.