Conservation of Mechanical Energy and Energy Principles

Conservation of Mechanical Energy

Definition and Principle

The conservation of mechanical energy states that if only conservative forces act on a system, the total mechanical energy (the sum of kinetic and potential energies) remains constant throughout the motion.

• Mechanical Energy (Emech): $E_{mech} = K + U$

• Conservative Forces: Forces for which the work done is independent of the path taken (e.g., gravity, spring force).

• Conservation Equation: $E_i = E_f$

• Expanded Form: $K_i + U_i = K_f + U_f$

• Change in Mechanical Energy: $\Delta E_{mech} = \Delta K + \Delta U = 0$

Example Application: Roller coaster problems, pendulum motion, and spring systems where friction and air resistance are negligible.

Problem Solving Using Conservation of Mechanical Energy

General Approach

When friction is absent, the speed and energy of an object depend only on its height and initial energy. The mechanical energy at different points can be calculated using the conservation principle.

• Step 1: Identify initial and final positions.

• Step 2: Write expressions for kinetic and potential energy at each position.

• Step 3: Set $E_i = E_f$ and solve for the unknown (velocity, height, etc.).

Example: For a roller coaster of mass 100 kg starting at rest at height $h_A = 40$ m:

• At point A: $E_A = mgh_A$

• At point B: $E_B = \frac{1}{2}mv_B^2 + mgh_B$

• At point C: $E_C = \frac{1}{2}mv_C^2 + mgh_C$

• Use $E_A = E_B = E_C$ to solve for $v_B$ and $v_C$.

Formula for velocity at a lower height: $v = \sqrt{2g(h_{initial} - h_{final})}$

Examples of Conservation of Mechanical Energy

Roller Coaster

Ignoring friction, the mechanical energy is conserved as the coaster moves along the track. The kinetic energy at a lower point is higher due to the conversion of potential energy to kinetic energy.

Toy Dart Gun

A dart of mass 0.100 kg is pressed against a spring (k = 250 N/m, x = 0.06 m) and released. The speed of the dart when the spring reaches its natural length is found by equating the initial elastic potential energy to the final kinetic energy:

• $E_i = E_f$

• $\frac{1}{2}kx^2 = \frac{1}{2}mv^2$

• $v = \sqrt{\frac{k}{m}}x$

• For the given values: $v = 3.0$ m/s

Pendulum Motion

At the highest point, all energy is potential; at the lowest, all is kinetic. At intermediate points, energy is shared between kinetic and potential forms.

Work Done by Nonconservative Forces

Nonconservative Forces and Energy Change

When nonconservative forces (e.g., friction, air resistance) act, mechanical energy is not conserved. These forces convert mechanical energy into other forms, such as heat or sound.

• Nonconservative Work: $W_{nc} = \Delta E = E_f - E_i$

• General Energy Equation: $W_{total} = W_c + W_{nc}$

• Expanded: $-\Delta U + W_{nc} = \Delta K$

• Nonconservative Forces: Friction, air resistance, tension

Example: A package slides into a spring and stops due to friction. The work done by friction equals the change in mechanical energy:

• $W_{friction} = -15d$

• $\frac{1}{2}mv_1^2 = \frac{1}{2}kd^2 + 15d$

• Solve for $d$ (compression distance)

Summary Table: Conservative vs. Nonconservative Forces

Type of Force	Examples	Work on Closed Path	Effect on Mechanical Energy
Conservative	Gravity, Spring	Zero	Conserves mechanical energy
Nonconservative	Friction, Air Resistance, Tension	Nonzero, depends on path	Converts mechanical energy to other forms

Key Formulas

• Gravitational Potential Energy: $U = mgh$

• Elastic Potential Energy (Spring): $U = \frac{1}{2}kx^2$

• Kinetic Energy: $K = \frac{1}{2}mv^2$

• Conservation of Mechanical Energy: $K_i + U_i = K_f + U_f$

• Work by Nonconservative Forces: $W_{nc} = \Delta K + \Delta U = \Delta E$

Summary of Chapter 7

• Conservative forces conserve mechanical energy.

• Nonconservative forces convert mechanical energy into other forms.

• Work done by a conservative force is independent of path and zero on closed paths.

• Work done by a nonconservative force depends on the path and is nonzero on closed paths.

• Energy in the form of potential energy can be converted to kinetic or other forms.

• Work done by a conservative force is the negative of the change in potential energy.

• Mechanical energy is conserved only in systems with purely conservative forces.

• Work done by nonconservative forces equals the change in a system's mechanical energy.