Work, Kinetic Energy and the Work–Energy Theorem

In this chapter we study three closely related ideas:

• Work – when a force acts and produces displacement.
• Kinetic energy – energy of motion.
• Work–Energy Theorem – work done = change in kinetic energy.

1. Work

In physics, a force does work only if it causes a body to move.

1.1 Work by a constant force in the direction of motion

Suppose a constant force \( F \) acts on a body, and it moves in a straight line a distance \( s \) in the same direction as the force. Then the work done is:

\[ W = F\,s \]

Units: Force in newton (N), distance in metre (m), so work is in joule (J): \( 1\ \text{J} = 1\ \text{N}\cdot\text{m} \).

1.2 Work when the force is at an angle

Often the force is not exactly along the direction of motion. Let:

• \( F \) = magnitude of the force,
• \( s \) = displacement,
• \( \theta \) = angle between the force and displacement.

Only the component of force along the displacement does work. The work done is:

\[ W = F\,s \cos\theta \]

1.3 Work done by a varying force

If the force changes from point to point, we consider a very small displacement \( d\mathbf{s} \). For that small step:

\[ dW = \mathbf{F} \cdot d\mathbf{s} \]

The total work from point 1 to point 2 is found by adding (integrating) these small contributions:

\[ W_{1 \to 2} = \int_{1}^{2} \mathbf{F} \cdot d\mathbf{s} \]

For one-dimensional motion along the \( x \)-axis:

\[ W_{1 \to 2} = \int_{x_1}^{x_2} F_x(x)\, dx \]

2. Kinetic Energy

A moving body has kinetic energy. For a particle of mass \( m \) moving with speed \( v \), the kinetic energy is:

\[ T = \frac{1}{2} m v^2 \]

• If speed becomes zero, kinetic energy is zero.
• If speed doubles, kinetic energy becomes four times (because \( v^2 \) term).

3. Work–Energy Theorem (Derivation in Simple Steps)

We now show that:

Work done on a particle = Change in its kinetic energy.
\[ W_{1 \to 2} = T_2 - T_1 \]

3.1 Starting from Newton’s second law

Newton’s second law for motion in a straight line:

\[ F = m a \]

Acceleration \( a \) can be written using the chain rule:

\[ a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v \frac{dv}{dx} \]

Substitute this into Newton’s law:

\[ F = m v \frac{dv}{dx} \]

3.2 Multiply both sides by \( dx \)

Multiply the above equation by \( dx \):

\[ F\,dx = m v\, dv \]

• Left side \( F\,dx \) is the small work done: \( dW \).
• Right side \( m v\,dv \) is a small change in something related to speed.

So we can write:

\[ dW = m v\, dv \]

3.3 Recognising kinetic energy

Note that:

\[ d\left(\frac{1}{2} m v^2\right) = m v\, dv \]

Thus:

\[ dW = d\left(\frac{1}{2} m v^2\right) = dT \]

Integrate between state 1 (speed \( v_1 \)) and state 2 (speed \( v_2 \)):

\[ \int_{1}^{2} dW = \int_{1}^{2} dT \quad \Rightarrow \quad W_{1 \to 2} = T_2 - T_1 \]

4. Conservative Forces and Potential Energy

A force is called conservative if the work done by it depends only on the starting and ending points, and not on the path taken.

• Gravitational force is conservative.
• Spring force is conservative.
• Friction is not conservative (work depends on path).

For a conservative force, we can define a potential energy \( V \) such that:

\[ \mathbf{F} = -\nabla V \]

4.1 Gravitational potential energy (near Earth)

If an object of mass \( m \) is at height \( h \) above the ground:

\[ V = m g h \]

When the object falls, \( h \) decreases, so \( V \) decreases and \( T \) increases.

4.2 Spring potential energy

For a spring of constant \( k \), extended or compressed by \( x \):

\[ V = \frac{1}{2} k x^2 \]

5. Conservation of Mechanical Energy

For conservative forces:

\[ W_{1 \to 2} = - (V_2 - V_1) = -\Delta V \]

Combine with work–energy theorem:

\[ T_2 - T_1 = - (V_2 - V_1) \]

Rearranging:

\[ T_1 + V_1 = T_2 + V_2 = \text{constant} \]

6. Power

Power is the rate at which work is done.

\[ P = \frac{dW}{dt} \]

Using \( dW = \mathbf{F} \cdot d\mathbf{s} \) and \( \mathbf{v} = \dfrac{d\mathbf{s}}{dt} \):

\[ P = \mathbf{F} \cdot \mathbf{v} \]

Units of power: watt (W) where \( 1\ \text{W} = 1\ \text{J/s} \).

Gravitational Potential Energy (GPE)

When you lift an object against gravity, you do work on it. This work gets stored as gravitational potential energy. The energy is stored because the object is now at a higher position in the gravitational field.

1. Definition

Gravitational potential energy is the energy a body possesses due to its height above a reference level (usually the ground).
The formula is:

\( U_g = mgh \)

• m = mass of body
• g = acceleration due to gravity
• h = height above reference point

2. Derivation From Work Done Against Gravity

The gravitational force on a body is:

\( F = mg \)

To lift the object slowly (no acceleration), we must apply an upward force of equal magnitude: \( F_{\text{applied}} = mg \).

Work done in raising the object through a small height \(dh\) is:

\( dW = F_{\text{applied}} \, dh = mg \, dh \)

Total work done from height 0 to height \(h\) is:

\( W = \int_0^h mg \, dh = mgh \)

This work is stored as gravitational potential energy:
\( \boxed{U_g = mgh} \)

3. Important Points

• GPE depends only on height, not on the path taken.
• GPE increases when the body goes higher.
• The reference point (where \(U_g = 0\)) can be chosen freely.

Elastic Potential Energy (EPE)

Elastic potential energy is the energy stored in a spring (or any elastic object) when it is stretched or compressed.

1. Hooke’s Law

For a spring obeying Hooke’s law:

\( F = -kx \)

• \(k\) = spring constant (“stiffness”)
• \(x\) = stretch or compression from natural length
• Negative sign means spring pulls opposite to deformation

2. Work Done in Stretching/Compressing a Spring

Work done on the spring is stored as elastic potential energy.
The work done in stretching the spring from 0 to \(x\) is:

\( W = \int_0^x kx' \, dx' = \frac{1}{2}kx^2 \)

So the elastic potential energy is:

\( \boxed{U_e = \frac{1}{2}kx^2} \)

3. Physical Meaning

• The spring stores energy when stretched or compressed.
• If released, this stored energy becomes kinetic energy.
• Stiffer springs (large \(k\)) store more energy for the same stretch.

4. Graphical Interpretation

The force on a spring increases linearly with extension (Hooke's law).
The work done (area under the \(F\)-\(x\) curve) is the area of a triangle.

Work \( = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}x \cdot kx = \frac{1}{2}kx^2 \)

Comparison Table

Conservative and Nonconservative Forces, Force–Potential Energy Relation, and Energy Diagrams

In mechanics, forces can be divided into two major categories: Conservative and Nonconservative forces. Understanding these is essential to study energy conservation.

1. Conservative Forces

A force is said to be conservative if the work it does on a particle moving between two points does not depend on the path taken, but only on the initial and final positions.

Examples:

• Gravitational force
• Spring (Hooke's law) force
• Electrostatic force

1.1 Defining properties of conservative forces

• Path independence:
  The work done from point A to B is the same for all paths.
• Zero net work in a closed loop:
  \[ W_{\text{closed loop}} = 0 \]
• Associated with potential energy:
  Conservative forces allow us to define a potential energy function \( V(x) \).

1.2 Work done by a conservative force

If a conservative force \( \vec{F} \) is acting along a path, the work done is:

\[ W = \int_{A}^{B} \vec{F} \cdot d\vec{r} \]

For a conservative force, this integral depends only on the end points A and B.

2. Nonconservative Forces

A force is nonconservative if the work done depends on the path taken and cannot be expressed as the difference of a potential energy.

Examples:

• Friction
• Air resistance (drag)
• Viscous forces

2.1 Properties

• Work depends on the path taken
• Work done in a closed loop is NOT zero:
  \[ W_{\text{closed loop}} \ne 0 \]
• No potential energy function can be associated with them
• They dissipate mechanical energy (usually as heat)

3. Force and Potential Energy

For conservative forces, there is a direct relationship between the force \(F(x)\) and the potential energy \(V(x)\).

3.1 One-Dimensional Case

For motion along x-axis:

\[ F(x) = -\frac{dV}{dx} \]

This tells us:

• Force is the negative slope of the potential energy curve.
• If \( V(x) \) increases with x, the force is negative (points left).
• If \( V(x) \) decreases with x, the force is positive (points right).

3.2 Derivation of \( F = -dV/dx \)

Work done by a conservative force:

\[ W = -\Delta V = -(V_B - V_A) \]

For a very small displacement \( dx \):

\[ dW = F dx \]

\[ dW = -dV \]

\[ F dx = -dV \quad\Rightarrow\quad F = -\frac{dV}{dx} \]

This is the fundamental connection between force and potential energy.

4. Energy Diagrams

Energy diagrams plot potential energy \(V(x)\) vs position \(x\). They help us visualize:

• Stable and unstable equilibrium points
• Turning points of motion
• Bound and unbound motion
• Allowed regions of motion

4.1 Features of an Energy Diagram

(a) Turning Points

Points where total energy \(E\) equals potential energy \(V(x)\):

\[ E = V(x) \]

At these points, kinetic energy becomes zero, and the particle reverses direction.

(b) Allowed and Forbidden Regions

Motion is allowed where:

\[ E > V(x) \]

Motion is forbidden (classically) where:

\[ E < V(x) \Rightarrow T < 0 \; \text{(impossible)} \]

(c) Equilibrium Points

Equilibrium occurs when:

\[ F = -\frac{dV}{dx} = 0 \]

\[ \frac{dV}{dx} = 0 \]

• Stable equilibrium: \(V(x)\) has a minimum
• Unstable equilibrium: \(V(x)\) has a maximum
• Neutral equilibrium: \(V(x)\) is flat

4.2 ASCII Diagram Examples

Stable Equilibrium (Potential Minimum):

   V(x)
    |            *
    |          *   *
    |        *       *
    |------*----x0----*---------
    |
    +---------------------------- x
         Stable Equilibrium

The particle tends to return to \(x_0\) if slightly displaced.

Unstable Equilibrium (Potential Maximum):

   V(x)
    |        *------*
    |      *          *
    |    *              *
    |---x0-------------------------
    |
    +---------------------------- x
       Unstable Equilibrium

The particle moves away from \(x_0\) if slightly displaced.

5. Mechanical Energy in Conservative Systems

If only conservative forces act:

\[ E = T + V = \text{constant} \]

This is the principle of conservation of mechanical energy.

6. Mechanical Energy in Nonconservative Systems

If nonconservative forces (like friction) act:

\[ W_{\text{nc}} = \Delta E = \Delta (T + V) \]

The total mechanical energy decreases because some energy is dissipated as heat or other forms.