Potential Energy, Thermal Energy, and Conservation of Energy

Section 10.4: Potential Energy

Potential Energy

Potential energy is stored energy that can be readily converted to other forms of energy, such as kinetic or thermal energy. The concept of potential energy is central to understanding how energy is conserved and transformed in physical systems.

• Conservative forces are forces that can store useful energy as potential energy. Examples include:

  • Gravity

  • Elastic forces (such as springs)

• Nonconservative forces (e.g., friction) cannot store useful energy as potential energy; instead, they dissipate energy, often as heat.

Gravitational Potential Energy

The change in gravitational potential energy is proportional to the change in an object's height. The gravitational potential energy of an object of mass m at height y is given by:

$$ U_g = mgy$$

• The reference level for potential energy can be chosen arbitrarily (e.g., $U_g = 0$ at $y = 0$).

• Only changes in $U_g$ matter for energy conservation.

• Because gravity is a conservative force, gravitational potential energy depends only on the height of the object, not the path taken to reach that height.

Example: Ranking Gravitational Potential Energy

Given several balls at different heights, their gravitational potential energies can be ranked from largest to smallest based on their vertical positions. The higher the ball, the greater its potential energy.

Example: Speed at the Bottom of a Hill

• For objects starting from rest and descending hills of the same height (regardless of steepness), the speed at the bottom is the same, assuming no friction. This is due to conservation of energy: all potential energy is converted to kinetic energy.

Example: Projectile Speeds

• Three balls thrown from a cliff with the same speed but at different angles will have the same speed just before hitting the ground, as energy conservation ensures the same total energy transformation.

Elastic Potential Energy

Elastic (or spring) potential energy is stored when a force compresses or stretches a spring. Hooke's law describes the force required to compress or stretch a spring:

$$ F = -kx$$

The elastic potential energy stored in a spring displaced a distance $x$ from equilibrium is:

$$ U_s = \frac{1}{2}kx^2$$

• Here, $k$ is the spring constant, and $x$ is the displacement from equilibrium.

Example: Energy in a Stretched Tendon

The Achilles tendon can be modeled as a spring. If the force increases linearly with extension, the spring constant $k$ is the slope of the force vs. extension graph. For a maximum stretch $x$:

$$ k = \frac{400\,\text{N}}{0.0024\,\text{m}} = 1.67 \times 10^5\,\text{N/m}$$U_s = \frac{1}{2}kx^2 = \frac{1}{2}(1.67 \times 10^5\,\text{N/m})(0.0050\,\text{m})^2 = 2.1\,\text{J}$$

This energy is a small but significant fraction of the kinetic energy involved in walking or running.

Section 10.5: Thermal Energy

Thermal Energy

Thermal energy is the sum of the kinetic energy of atoms and molecules in a substance and the elastic potential energy stored in the molecular bonds between atoms.

Creating Thermal Energy

• Friction on a moving object does work, converting mechanical energy into thermal energy:

$$ \Delta E_{th} = f_k \Delta x$$

• Drag also does work and creates thermal energy:

$$ \Delta E_{th} = D \Delta x$$

• The work-energy equation, including thermal energy, is:

$$ \Delta K + \Delta U + \Delta E_{th} = W$$

• Try It Yourself: Rubbing a soft object on a surface generates thermal energy, which can be felt as warmth due to the increased motion of atoms and molecules.

Section 10.6: Conservation of Energy

Conservation of Energy

The law of conservation of energy states that the total energy of an isolated system remains constant. Energy can be transformed from one form to another, but it cannot be created or destroyed.

• The general conservation of energy equation is:

$$ \Delta K + \Delta U + \Delta E_{th} = W$$

• For an isolated system ($W = 0$):

$$ K_i + U_i + \Delta E_{th,i} = K_f + U_f + \Delta E_{th,f}$$

• This approach is analogous to the before-and-after method used in conservation of momentum.

Choosing an Isolated System

When applying conservation of energy, it is important to define the system boundaries. The system may include:

• An object in free fall (object + Earth)

• An object sliding down a frictionless ramp (object + Earth)

• An object compressing a spring (object + spring)

• An object sliding along a surface with friction (object + surface)

Example Problems

Example: Car Rolling Down a Hill

• System: The car

• Initial state: At rest at the top of the hill

• Final state: After descending 5.0 m

• Energy transformation: Gravitational potential energy to kinetic energy

$$ K_i + U_{g,i} = K_f + U_{g,f}$$

With $K_i = 0$ and $U_{g,f} = 0$:

$$ \frac{1}{2}mv^2 = mgh$$v = \sqrt{2gh}$$

Example: Child Sliding Down a Slide

• System: The child

• Constant speed implies work is done by friction (thermal energy generated)

• Energy transformation: Gravitational potential energy to thermal energy

$$ K_i + U_{g,i} + \Delta E_{th} = K_f + U_{g,f}$$

Example: Speed of a Spring-Launched Ball

• System: Ball + spring

• Initial state: Spring compressed, ball at rest

• Final state: Spring at equilibrium, ball in motion

• Energy transformation: Elastic potential energy to kinetic energy

$$ K_i + U_{s,i} = K_f + U_{s,f}$$

With $K_i = 0$, $U_{s,f} = 0$:

$$ \frac{1}{2}mv^2 = \frac{1}{2}kx^2$$v = \sqrt{\frac{kx^2}{m}}$$

Example: Thermal Energy of a Trip Down a Slide

• System: Quinn + Earth + slide

• Initial state: Quinn at rest at the top

• Final state: Quinn moving at the bottom

• Energy transformation: Gravitational potential energy to kinetic and thermal energy

$$ \Delta E_{th} = (U_{g,i}) - K_f - (U_{g,f})$$

For a slide of length 5.0 m at 30°, $h = 2.5$ m, $m = 24$ kg, $v = 2.0$ m/s:

$$ \Delta E_{th} = mgh - \frac{1}{2}mv^2 = 588\,\text{J} - 48\,\text{J} = 540\,\text{J}$$

Summary Table: Forms of Energy and Conservation

Form of Energy	Expression	Description
Kinetic Energy	$K = \frac{1}{2}mv^2$	Energy of motion
Gravitational Potential Energy	$U_g = mgy$	Energy due to position in a gravitational field
Elastic Potential Energy	$U_s = \frac{1}{2}kx^2$	Energy stored in a stretched or compressed spring
Thermal Energy	--	Sum of kinetic and potential energies of atoms/molecules

Key Equations

• Work done by a force: $W = F_{ext} d$

• Conservation of energy (general): $K_i + U_i + \Delta E_{th} = K_f + U_f + W$

• For an isolated system: $K_i + U_i + \Delta E_{th} = K_f + U_f$

Additional info:

• All QuickCheck questions reinforce the principle that, for conservative forces, only the initial and final positions (not the path or angle) affect the energy transformations.

• Thermal energy is always generated when nonconservative forces (like friction) are present.