Work and Energy

Internal forces do no work on a rigid body because:

• According to Newton's Third Law, internal forces occur in equal and opposite pairs
• In a rigid body, the distances between particles remain fixed
• When the body moves, the work done by one force of a pair exactly cancels the work done by its partner
• Mathematically: $W = \int (\vec{F}_{AB} \cdot d\vec{r}_A + \vec{F}_{BA} \cdot d\vec{r}_B) = 0$ since $\vec{F}_{AB} = -\vec{F}_{BA}$ and $d\vec{r}_A = d\vec{r}_B$ in rigid motion

Effects on potential energy:

• Potential energy changes only when the separation between interacting parts changes
• For a rigid body, internal potential energy remains constant during motion
• We only need to consider potential energy associated with external force fields (like gravity)
• When analyzing rigid body systems, we can focus on the body's overall position rather than internal configurations

Work done by the spring force in different transitions:

1. Natural → Compressed (displacement 0 to -x): $W = -\frac{1}{2}kx^2$ (negative work) Spring force opposes compression

2. Natural → Extended (displacement 0 to +x): $W = -\frac{1}{2}kx^2$ (negative work) Spring force opposes extension

3. Compressed → Natural (displacement -x to 0): $W = +\frac{1}{2}kx^2$ (positive work) Spring force assists the motion

4. Extended → Natural (displacement +x to 0): $W = +\frac{1}{2}kx^2$ (positive work) Spring force assists the motion

Sign convention explanation:

• Negative work: Spring force opposes displacement (energy is stored in spring)
• Positive work: Spring force assists displacement (spring releases stored energy)
• In a complete cycle, net work done by spring force is zero (conservative force)

Note: The elastic potential energy change is always $\Delta U = \frac{1}{2}kx_f^2 - \frac{1}{2}kx_i^2$

The spring force performs zero net work during a complete cycle of motion when:

1. The block returns to its starting position (completes a round trip)
2. The displacement follows any closed path

Key principles:

• The spring force is conservative
• Work done by conservative forces depends only on initial and final positions, not the path taken
• During extension: W₁ = -½kx²
• During compression (return): W₂ = +½kx²
• Total work: W₁ + W₂ = 0

This is a defining characteristic of conservative forces - they perform zero work over any closed path, allowing for the definition of potential energy.

Gravity is conservative because:

• Work done by gravity in a closed path (round trip) equals zero
• Work depends only on initial and final positions, not the path taken
• For any path between points A and B, W = mgh (where h is vertical distance)

This demonstrates that:

• Gravitational potential energy is a state function (depends only on position)
• Energy is conserved in gravitational systems without non-conservative forces
• A complete height-to-height return cycle results in zero net work

Example: A roller coaster on a frictionless track would theoretically return to the same height with the same speed, regardless of the track shape between those points

The work done equals zero in these transitions because:

• When moving from elongated (x) to compressed (-x), the path must pass through the natural length (0)
• The complete path can be broken into two segments:
  1. Elongated to natural: Work = +½kx² (positive)
  2. Natural to compressed: Work = -½kx² (negative)
• The total work is the sum: +½kx² + (-½kx²) = 0

This illustrates the path-dependent nature of work for non-conservative forces, where the net work in a round trip equals zero. The spring force is conservative, meaning the work done depends only on initial and final positions, not the path taken.

Potential energy emerges from particle motion by observing that:

1. When particles in a conservative system move apart, their kinetic energy decreases
2. This "lost" kinetic energy isn't truly lost but stored in another form
3. When particles return to their original configuration, the kinetic energy returns
4. Therefore, we define potential energy as energy stored in the configuration of a system

This leads to the key insight that:

• Potential energy is configuration-dependent rather than motion-dependent
• It represents the capacity to do work based on position/arrangement
• It allows us to maintain the principle of energy conservation
• Changes in potential energy equal negative work done by conservative forces

Example: When a pendulum swings, kinetic energy at the bottom transforms to potential energy at the maximum height, then back to kinetic energy.

For a block sliding in a conservative system (no friction or other dissipative forces):

1. Total mechanical energy remains constant: $E_{total} = K + U = constant$

2. Energy conversion occurs between:

   • Gravitational potential energy: $U = mgh$
   • Kinetic energy: $K = \frac{1}{2}mv^2$

3. Key equation: $\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f$

4. For motion from rest at height h to a lower position:

   • Initial state: $v_i = 0, h_i = h$
   • At any point: $\frac{1}{2}mv^2 = mg(h - h')$ where h' is current height

This analysis works for any path shape, as long as non-conservative forces are absent. The motion depends only on the change in height, not the specific path taken between points.

The potential energy increases by mgl(1-cosθ) compared to the lowest position, where:

• m is the mass of the pendulum bob
• g is the acceleration due to gravity
• l is the length of the pendulum
• θ is the angle from vertical

This represents the gravitational potential energy gained as the bob rises to height l(1-cosθ) above its lowest position.

The descent height h relates to the initial velocity v of the mass by: $h = v\sqrt{m/k}$

Where:

• m is the mass of the object
• k is the spring constant
• v is the initial velocity

This formula applies when considering a mass-spring system in a gravitational field, though the same result is obtained if gravity is neglected and the equilibrium position is taken as the natural length.

Mass-energy equivalence: $E = mc^2$

Where:

• E is energy
• m is mass
• c is speed of light in vacuum (3×10⁸ m/s)

For an electron:

• Energy equivalent ≈ 8.18×10⁻¹⁴ J
• This is the energy released if an electron is completely converted to energy
• The same amount of energy would be required to create an electron

This relationship demonstrates that mass itself is a form of energy.