Conservation of Energy & Problem Solving in Physics

Conservation of Energy: Principles and Problem Solving

Introduction to Energy Conservation

The conservation of energy is a fundamental concept in physics, stating that the total energy of an isolated system remains constant unless acted upon by external forces. This principle is widely used to analyze mechanical systems, especially in the context of kinetic and potential energy transformations.

• System Definition: Clearly define the objects included in your system.

• Open vs. Closed Systems:

  • Open System: Energy can be transferred in or out (e.g., via friction, air resistance, applied forces, or thrust).

  • Closed System: No energy is transferred in or out; only internal forces act.

• Non-conservative Work: Work done by non-conservative forces (e.g., friction, air resistance) must be included as energy losses or gains ($W_{NC}$ or $W_{air}$).

Energy Accounting Table

To systematically solve energy problems, use an energy accounting table to track different forms of energy at key moments ("snapshots") in the process.

Snapshot	KE (Kinetic Energy)	PEgrav (Gravitational Potential Energy)	PEspring (Spring Potential Energy)	WNC (Work by Non-conservative Forces)
#1 (Initial)	Depends on problem	Depends on height	Depends on compression/stretch	External energy added?
#2 (Final)	Depends on problem	Depends on height	Depends on compression/stretch	External energy lost?

Energy Conservation Equation:

$KE_1 + PE_{grav,1} + PE_{spring,1} + W_{NC,1} = KE_2 + PE_{grav,2} + PE_{spring,2} + W_{NC,2}$

For a closed system with no non-conservative work:

$KE_1 + PE_{grav,1} + PE_{spring,1} = KE_2 + PE_{grav,2} + PE_{spring,2}$

Projectile Motion and Energy Conservation

Projectile Launched Horizontally from a Height

When a projectile of mass m is launched horizontally with speed $v_0$ from a height h, its speed upon reaching the ground can be found using energy conservation.

• Kinetic Energy (KE): $KE = \frac{1}{2}mv^2$

• Gravitational Potential Energy (PEgrav): $PE_{grav} = mgh$

• Conservation Equation: $\frac{1}{2}mv_0^2 + mgh = \frac{1}{2}mv^2$

• Solve for final speed: $v = \sqrt{v_0^2 + 2gh}$

Example: A projectile is launched horizontally at $v_0$ from height $h$. Its speed when it hits the ground is $v = \sqrt{v_0^2 + 2gh}$.

Projectile Launched at an Angle

For a projectile launched from the ground at angle $\theta$ with speed $v_0$:

• At the top of the arc: Only the horizontal component of velocity remains ($v_{top} = v_0 \cos\theta$).

• When returning to the ground: If air resistance is neglected, the speed is the same as the initial speed ($v = v_0$).

Example: A projectile launched at $v_0$ returns to the ground with speed $v_0$ (neglecting air resistance).

Energy Losses: Air Resistance and Friction

Projectile with Air Resistance

When air resistance is present, some mechanical energy is lost to the environment. The difference between the expected (no-loss) and actual kinetic energy gives the energy lost.

• Energy Lost to Air Resistance: $E_{lost} = (\text{Initial Total Energy}) - (\text{Final Total Energy})$

• Example Calculation: If a 1 kg projectile is launched at 10 m/s from a 3 m cliff and lands at 5 m/s, the energy lost is:

$\frac{1}{2}mv_0^2 + mgh = \frac{1}{2}mv^2 + E_{lost}$ $50\,J + 29.4\,J = 12.5\,J + E_{lost}$ $E_{lost} = 66.9\,J$

Block Sliding Down a Ramp: Frictionless and With Friction

When a block slides down a ramp, gravitational potential energy is converted to kinetic energy. If friction is present, some energy is lost as heat.

• Frictionless Ramp: $mgh = \frac{1}{2}mv^2$

• With Friction: $mgh = \frac{1}{2}mv^2 + \mu mg\Delta x \cos\theta$

• Solving for Coefficient of Friction ($\mu$): $\mu = \frac{gh - \frac{1}{2}v^2}{g\Delta x \cos\theta}$

Example: A 2 kg block slides down a 1 m ramp at $30^\circ$. If final speed is less than expected, use the above formula to find $\mu$.

Summary Table: Energy Forms and Problem-Solving Steps

Energy Form	Expression	When Used
Kinetic Energy (KE)	$\frac{1}{2}mv^2$	Object in motion
Gravitational Potential Energy (PEgrav)	$mgh$	Object at height $h$
Spring Potential Energy (PEspring)	$\frac{1}{2}kx^2$	Compressed or stretched spring
Work by Non-conservative Forces ($W_{NC}$)	Varies (e.g., $F_{fric} \Delta x$)	Friction, air resistance, applied forces

Problem-Solving Steps

1. Define your system and identify if it is open or closed.

2. List all forms of energy present at each key moment (snapshot).

3. Write the energy conservation equation, including work by non-conservative forces if present.

4. Solve for the unknown quantity (e.g., final speed, energy lost, coefficient of friction).

Additional info: The notes also emphasize the importance of clearly identifying energy gains and losses, and using systematic tables to organize energy terms for each problem scenario.