9.2 Mechanical Energy and Conservation of Energy

Mechanical energy describes the energy an object has because of its motion or its position. It comes in two forms: kinetic energy (from movement) and potential energy (stored). Understanding how these two forms trade off is central to solving problems in mechanics.

Conservation of energy is one of the most powerful principles in physics. Energy can't be created or destroyed, only transformed from one form to another. In a closed system, the total energy stays constant, which gives you a reliable equation to work with even when forces and accelerations get complicated.

Mechanical Energy

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Changes in kinetic and potential energy

Kinetic energy ($KE$) is the energy an object has because it's moving:

$KE = \frac{1}{2}mv^2$

where $m$ is mass (kg) and $v$ is velocity (m/s). Notice that velocity is squared, so doubling your speed quadruples your kinetic energy. A 1,200 kg car traveling at 20 m/s has $KE = \frac{1}{2}(1200)(20)^2 = 240{,}000 \text{ J}$, but at 40 m/s that jumps to 960,000 J.

Potential energy ($PE$) is energy stored due to an object's position or configuration. There are two main types you'll work with:

• Gravitational potential energy: $PE_g = mgh$, where $g$ is gravitational acceleration (9.8 m/s²) and $h$ is height relative to a chosen reference point. A 2 kg book sitting 1.5 m above the floor has $PE_g = (2)(9.8)(1.5) = 29.4 \text{ J}$ relative to the floor. Your choice of reference point matters for the numbers, but differences in height are what actually affect the physics.
• Elastic potential energy: $PE_e = \frac{1}{2}kx^2$, where $k$ is the spring constant (N/m) and $x$ is the displacement from the spring's natural (equilibrium) length. A spring with $k = 500$ N/m compressed 0.1 m stores $PE_e = \frac{1}{2}(500)(0.1)^2 = 2.5 \text{ J}$.

To find how much energy changes between two states, use these:

• $\Delta KE = \frac{1}{2}m(v_f^2 - v_i^2)$
• $\Delta PE_g = mg(h_f - h_i)$
• $\Delta PE_e = \frac{1}{2}k(x_f^2 - x_i^2)$

where subscripts $i$ and $f$ stand for initial and final values.

Conservation of Energy

Conservation of energy in mechanics

The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In a closed system, total energy remains constant.

The classic example is a pendulum. At the top of its swing, the pendulum is momentarily at rest (maximum $PE_g$, zero $KE$). At the bottom of its swing, it moves fastest (maximum $KE$, minimum $PE_g$). Energy continuously shifts between these two forms.

When no non-conservative forces (like friction or air resistance) act on a system, mechanical energy is conserved:

$KE_i + PE_i = KE_f + PE_f$

This is the equation you'll use most often. On a frictionless roller coaster, for instance, the car's total mechanical energy at the top of the first hill equals its total mechanical energy at every other point on the track. If the car starts from rest at a height of 30 m, you can find its speed at any other height without knowing anything about the shape of the track.

Work-energy relationships in physics

Work ($W$) is the transfer of energy to an object by a force acting over a displacement:

$W = Fd\cos\theta$

where $F$ is the magnitude of the force, $d$ is the displacement, and $\theta$ is the angle between the force and displacement vectors. When the force is in the same direction as the motion, $\theta = 0°$ and $\cos\theta = 1$, so $W = Fd$. When the force is perpendicular to the motion (like a normal force on a flat surface), $\theta = 90°$ and $W = 0$.

The work-energy theorem connects work directly to kinetic energy:

$W_{net} = \Delta KE = \frac{1}{2}m(v_f^2 - v_i^2)$

The net work done on an object equals the change in its kinetic energy. If positive net work is done, the object speeds up. If negative net work is done, it slows down.

Work done by conservative forces can be expressed in terms of potential energy changes:

1. Gravitational force: $W_g = -\Delta PE_g = -mg(h_f - h_i)$

2. Spring force: $W_s = -\Delta PE_e = -\frac{1}{2}k(x_f^2 - x_i^2)$

The negative sign is important. When an object falls ($h_f < h_i$), gravity does positive work and potential energy decreases. The signs are consistent: the energy that leaves the potential energy "account" shows up as kinetic energy.

Work done by non-conservative forces removes mechanical energy from the system. For kinetic friction:

$W_f = -f_k d$

where $f_k$ is the kinetic friction force and $d$ is the distance traveled. This is always negative because friction opposes motion. The "lost" mechanical energy becomes thermal energy. When friction is present, the full energy equation becomes:

$KE_i + PE_i + W_{nc} = KE_f + PE_f$

where $W_{nc}$ is the total work done by all non-conservative forces.

Power ($P$) measures how quickly work is done:

$P = \frac{W}{\Delta t}$

The unit is the watt (W), where 1 W = 1 J/s. A 200 W motor and a 100 W motor can both do 10,000 J of work, but the 200 W motor does it in half the time.

Energy in Systems and Equilibrium

A closed system is one where no energy enters or leaves. Energy can shift between forms inside the system (kinetic to potential, mechanical to thermal), but the total stays the same.

Non-conservative forces like friction and air resistance convert organized mechanical energy into disorganized thermal energy. That thermal energy is still energy, so total energy is conserved, but the mechanical energy available to do useful work decreases.

Mechanical equilibrium occurs when the net force on an object is zero. At equilibrium, there's no acceleration, so kinetic energy isn't changing. For a spring, equilibrium is the natural length position. For gravitational situations, equilibrium is where the net force balances out (like a ball resting at the bottom of a bowl).