Chapter 8: Conservation of Energy

Introduction to Key Concepts
- Conservative and Nonconservative Forces
- Potential Energy
- Mechanical Energy and Its Conservation
- Problem Solving Using Conservation of Mechanical Energy
- The Law of Conservation of Energy
- Energy Conservation with Dissipative Forces: Solving Problems
- Gravitational Potential Energy and Escape Velocity
- Power
- Potential Energy Diagrams; Stable and Unstable Equilibrium
- Gravitational Assist (Gravitational Slingshot)

A conservative force is defined as follows:
- The work done by the force on an object moving from one point to another depends only on the initial and final positions of the object, independent of the path taken.
- Example of a conservative force: Gravity.
Another definition of a conservative force states:
- The net work done on an object moving around any closed path is zero.
Nonconservative forces example:
- Friction is a nonconservative force because the work done depends on both the starting and ending points and also on the path taken.
Potential energy can only be defined for conservative forces, highlighting the distinction between work done by conservative and nonconservative forces.
The work done by nonconservative forces equals the total change in kinetic and potential energies, formalized as:
- W_{NC} = riangle K + riangle U

Potential Energy is defined as energy associated with the forces depending on the position or configuration of an object (or objects) relative to its surroundings.
Familiar examples of potential energy include:
- A wound-up spring
- A stretched elastic band
- An object at a height above the ground
In raising a mass m to a height h, the work done by the external force is equated to the gravitational potential energy:
- U_{grav} = mgy

A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3:
- (a) Calculate the gravitational potential energy at points 2 and 3 relative to point 1, with y = 0 at point 1.
- (b) Determine the change in potential energy when the car moves from point 2 to point 3.
- (c) Repeat parts (a) and (b) with y = 0 at point 3.
Potential energy can convert to kinetic energy if the object is dropped.
Potential energy is a characteristic of a system, depending on external forces; thus, determining a reference point for measurement does not affect consistency as long as y = 0 is chosen uniformly.
General definition of gravitational potential energy:
- U_{grav} = mgy

If nonconservative forces are absent, the total mechanical energy remains constant:
- The sum of the changes in kinetic energy and potential energy is zero:
- riangle K + riangle U = 0
This leads to the definition of total mechanical energy:
- E = K + U
- Where K is kinetic energy and U is potential energy.
The principle of conservation of mechanical energy states that if only conservative forces do work, the total mechanical energy remains constant throughout a process.
- K1 + U1 = K2 + U2

At any point, the total mechanical energy is represented as:
- E = K + U = rac{1}{2} mv^2 + mgy
For example, consider a rock with an initial height y1 = h = 3.0 m. To find the velocity when it has fallen to 1.0 m above the ground, apply:
- rac{1}{2} mv1^2 + mgy1 = rac{1}{2} mv2^2 + mgy2

Determine the speed of a roller-coaster car based on height compared to starting height; analyze two water slides with different shapes but equivalent starting heights.
Consider specific queries about which rider will reach the bottom faster under ideal conditions (ignoring friction).

Address the circumstances when a roller-coaster car reaches a height of only 25 m on a second hill after traveling 400 m. Calculate thermal energy produced and estimate average friction force (assuming mass = 1000 kg).

Nonconservative forces, such as friction, do not conserve mechanical energy; the work done converts mechanical energy into other forms like thermal, electrical, or chemical energy.
The fundamental principle of the law of conservation of energy is:
- The total energy is neither gained nor lost in any process; energy can transform and transfer forms but remains constant.
- Expressed mathematically as:
- riangle K + riangle U + [ ext{change in all other forms of energy}] = 0

The force of gravity varies outside Earth’s surface. The work done moving within Earth’s gravitational field is given by:
- W = rac{GmM}{r}
- Upon solving, the gravitational potential energy is defined as:
- U(r) = - rac{GmM}{r}

To estimate speed just before impact after being released from a rocket at 1600 km altitude traveling 1800 m/s, disregard air resistance and calculate.

Escape velocity is the speed at which an object’s initial kinetic energy equals the magnitude of its potential energy at Earth’s surface, leading to a total energy of zero.
- For Earth, the escape velocity can be calculated using known values:
- v_{escape} = ext{[calculation formula]}

In comparing escape velocities for rockets on both Earth and Moon, utilize the following data:
- Moon mass: M_M = 7.35 imes 10^{22} ext{ kg}
- Moon radius: r_M = 1.74 imes 10^6 ext{ m}
- Earth mass: M_E = 5.98 imes 10^{24} ext{ kg}
- Earth radius: r_E = 6.38 imes 10^6 ext{ m}

Power is defined as the rate at which work is done, with SI unit measured in watts.
Power can also be expressed as the rate of energy transformation. It depends on force and velocity:
- P = rac{W}{t}
- P = rac{dW}{dt}
- P = rac{dE}{dt}

A 60-kg jogger runs up 4.5 m of stairs in 4.0 s:
- (a) Estimate power output in watts and horsepower.
- (b) Calculate the energy expended.

To find power requirements for a 1400-kg car:
- (a) While climbing a 10° hill at 80 km/h;
- (b) When accelerating from 90 km/h to 110 km/h in 6.0 s with an average resisting force of 700 N.

This section involves potential energy diagrams for particles affected by conservative forces.
- The object's behavior depends on its total energy.
- Objects oscillating within certain energy levels (E1, E2) have defined turning points; an energy of E0 denotes stable equilibrium, while energy at x4 leads to unstable equilibrium.

Spacecraft can gain speed from flying near a planet or moon, minimizing required fuel and energy for launch.
The change in energy during this maneuver is quantifiable through derived formulas.

Conservative forces yield results based only on initial and final positions.
Potential energy accounts for position-dependent forces.
Total mechanical energy is the sum of kinetic and potential energies.
When nonconservative forces are present, total energy, including all forms, is conserved.