6 - Work, Energy, and Conservation of Energy

Work from Constant and Variable Forces

The central idea is that forces can transfer energy through displacement. For a constant force, the work done is

Here, $\theta$ is the angle between the force and displacement.

• Positive work occurs when the force has a component along the displacement.

• Negative work occurs when the force has a component opposite the displacement.

• Zero work occurs when the force is perpendicular to the displacement.

For example, a horizontal force of $20\,\text{N}$ pushing a box through $3.0\,\text{m}$ in the same direction does

Kinetic friction of magnitude $8.0\,\text{N}$ over the same distance does

Takeaway: Always compare the force direction with the displacement direction before assigning the sign of work. Work and energy are measured in joules, where $1\,\text{J}=1\,\text{N}\cdot\text{m}$.

Work by a Variable Force

When a force changes in magnitude or direction, divide the path into small displacement elements. The work over the full path is

For one-dimensional motion along the $x$-axis,

This integral is the signed area under a force–position graph. Area above the horizontal axis contributes positive work; area below the axis contributes negative work.

For an ideal spring, Hooke’s law gives

The work done by the spring as it moves from $x_A$ to $x_B$ is

The spring force points toward equilibrium. Therefore, a spring released from a stretch or compression does positive work while moving toward equilibrium.

Takeaway: Use the constant-force formula only when the force is constant over the relevant displacement. Otherwise, use an integral or the signed area under the force–position graph.

Kinetic Energy and the Work–Energy Theorem

Kinetic energy describes motion:

Because speed is squared, changing speed can produce a larger change in energy. For example,

The work–energy theorem connects force information to motion:

Using the kinetic-energy formula,

Thus:

• If $W_{\text{net}}>0$, the object speeds up.

• If $W_{\text{net}}<0$, the object slows down.

• If $W_{\text{net}}=0$, the speed is unchanged, though the direction may change.

For a $2.0\,\text{kg}$ cart initially moving at $3.0\,\text{m/s}$ and experiencing $18\,\text{J}$ of net work,

Solving gives $v_f^2=27$, so $v_f\approx 5.2\,\text{m/s}$.

Takeaway: When initial speed, final speed, or net work is central to the problem, start with the work–energy theorem.

Potential Energy and Conservative Forces

Potential energy is useful for conservative forces. A conservative force has work that depends only on the initial and final positions, and its work around a closed path is zero. Its work is related to potential energy by

Near Earth’s surface, gravitational potential energy is

Gravity does work

For an ideal spring, elastic potential energy is

This quantity is positive for both a stretch and a compression because $x^2$ is nonnegative.

The zero level for potential energy is arbitrary. Choose a reference height or position that simplifies the equations, then use changes consistently.

Takeaway: A decrease in potential energy corresponds to positive work by the associated conservative force; an increase corresponds to negative work.

Conservation of Mechanical Energy

Mechanical energy combines kinetic and potential energy:

When only conservative forces do work, mechanical energy is conserved:

For an object moving vertically near Earth with no air resistance,

For an object released from rest at height $h$, with the ground chosen as $y=0$,

The mass cancels, giving

Therefore, the final speed depends on the drop height rather than the object’s mass, provided air resistance is negligible.

Takeaway: Use mechanical-energy conservation when the initial and final positions and speeds are easier to describe than the detailed motion between them.

Nonconservative Work and Energy Transfer

Friction, air resistance, and some applied forces are nonconservative. They can transfer mechanical energy into thermal energy or other forms. The appropriate equation is

Equivalently,

If kinetic friction does negative work, mechanical energy decreases by the magnitude of that work. For a block with initial kinetic energy $50\,\text{J}$ on a level surface, if friction does $-18\,\text{J}$ of work, then

The missing $18\,\text{J}$ of mechanical energy becomes thermal energy in the block and surface. Total energy remains conserved when the system includes the relevant surroundings.

Takeaway: Do not set initial and final mechanical energy equal when friction or another nonconservative force does work; include its work explicitly.

Choosing and Checking an Energy Method

A reliable solution begins by defining the system and the initial and final states.

1. Choose the system and decide which objects are included.

2. Record the initial and final speeds, heights, spring displacements, and other relevant states.

3. Choose a convenient reference level for gravitational potential energy.

4. Identify every force that does work and determine the sign of each contribution.

5. Use the work–energy theorem when forces, work values, or a force–position graph are given.

6. Use mechanical-energy conservation when only conservative forces do work.

7. Use $K_i+U_i+W_{\text{nc}}=K_f+U_f$ when friction, air resistance, or another nonconservative force does work.

8. Write the symbolic equation before substituting numerical values.

9. Check units: work, kinetic energy, potential energy, and mechanical energy should be in joules.

10. Check whether the sign and result match the physical situation.

Common errors include treating work as a vector, omitting $\cos\theta$, assuming every force does work, forgetting the negative work of friction, and treating mechanical energy as conserved despite nonconservative work.

Final takeaway: Select the method based on the information available: work–energy for net work and speed changes, mechanical-energy conservation for conservative motion, and the nonconservative-work equation when mechanical energy is transferred to other forms.