Conservation of Energy

Work, Energy, and Power

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In mechanics, this often involves kinetic energy, potential energy, and work done by forces.

• Kinetic Energy (KE): The energy of motion, given by .

• Potential Energy (PE): The energy stored due to position, such as gravitational potential energy .

• Work (W): The process of energy transfer, .

• Power (P): The rate of doing work, .

Example: Dragging a child on a carpet involves calculating the work done against friction and the change in energy. The total work includes both the force applied and the energy required to lift the carpet.

Application: Running down a mountain involves converting gravitational potential energy into kinetic energy and work done against friction.

• Energy Conservation Equation:

Energy in Roller Coasters and Ramps

Roller coaster problems require tracking energy transformations between kinetic and potential energy, and accounting for work done by friction.

• Mechanical Energy:

• Work by Friction:

Example: Calculating the minimum force required to move a roller coaster car up a hill, considering energy lost to friction.

Linear Momentum and Impulse

Momentum Conservation

Linear momentum is a measure of an object's motion, defined as . The law of conservation of momentum states that the total momentum of a closed system remains constant unless acted upon by external forces.

• Impulse (J): The change in momentum, .

• Collisions: In elastic and inelastic collisions, momentum is conserved.

Example: Calculating the total momentum of a system of moving objects, or the impulse delivered during a collision.

Applications of Momentum

• Projectile Motion: The change in momentum of a ball thrown vertically and its impact with the ground.

• Explosions: Conservation of momentum applies when an object explodes into fragments, with the total momentum before and after the explosion being equal.

• Collisions in Sports: Calculating the speed of a golf ball after being struck, or the time of contact during a collision.

Formulas:

Torque and Angular Momentum

Torque and Rotational Equilibrium

Torque is the rotational equivalent of force, causing objects to rotate about an axis. It is defined as , where is the lever arm and is the force applied.

• Rotational Equilibrium: An object is in rotational equilibrium if the sum of all torques acting on it is zero.

• Applications: Calculating the force required to lift a child on a seesaw, or the tension in a wire supporting a sign.

Example: Determining the torque required for a construction worker standing on a beam, or the muscle force needed to hold a weight with an outstretched arm.

• For equilibrium:

Angular Momentum

Angular momentum is a measure of rotational motion, given by , where is the moment of inertia and is the angular velocity.

• Conservation of Angular Momentum: In the absence of external torques, angular momentum is conserved.

Sample Calculations and Problem Types

Energy and Power Problems

• Calculating work done against friction and lifting objects

• Determining power output required for a moving vehicle

Momentum and Impulse Problems

• Finding total momentum of a system

• Calculating impulse delivered during collisions

• Determining final velocities after collisions

Torque and Rotational Equilibrium Problems

• Calculating torque for seesaws, beams, and signs

• Finding forces required for equilibrium

Key Equations Summary

Additional info:

• Problems cover topics from Ch 06 (Conservation of Energy), Ch 07 (Linear Momentum), and Ch 08 (Torque and Angular Momentum).

• Some questions involve real-world applications such as sports, playground equipment, and construction scenarios.

• Diagrams and calculations are used to illustrate physical principles and problem-solving strategies.