Work and Kinetic Energy

Kinetic Energy

Kinetic energy is a fundamental concept in physics, representing the energy an object possesses due to its motion. It is a scalar quantity and shares the same units as work, namely joules (J) in the SI system.

• Definition: The kinetic energy (KE) of an object with mass m and speed v is given by: SI Unit: 1 joule (J) = 1 kg·m2/s2

• Key Properties:

  • Kinetic energy increases with both mass and the square of speed.

  • For two objects with the same mass, doubling the speed increases kinetic energy by a factor of four.

• Example: A small airplane of mass 690 kg with kinetic energy 25,000 J has a speed m/s. If the speed is tripled, kinetic energy increases by a factor of nine.

Comparing Kinetic Energies

When comparing objects, both mass and speed affect kinetic energy. For example, if Ball A has half the mass and eight times the kinetic energy of Ball B, the speed ratio is 4.

Work and the Work-Energy Theorem

Definition of Work

Work is the process of energy transfer to or from an object via the application of force along a displacement. For a constant force:

• Units: Joules (J) or Newton-meters (N·m)

Work-Energy Theorem

The work-energy theorem states that the net work done by all forces acting on a particle equals the change in the particle's kinetic energy:

• If , kinetic energy increases (object speeds up).

• If , kinetic energy decreases (object slows down).

• If , kinetic energy remains constant (constant speed).

Application: Calculating Work and Kinetic Energy

• Example 1: Accelerating a 1000-kg car from 20 m/s to 30 m/s: J = 250 kJ

• Quick Check: For boxes pulled across a frictionless floor, the box with the largest force experiences the greatest change in kinetic energy, since .

Work Done by a Constant Force

When a constant force acts over a displacement, the work done is the product of the force and the displacement in the direction of the force:

• (if force and displacement are parallel)

• Graphically, work is the area under the force vs. displacement curve.

Work Done by a Varying Force

For forces that change with position, work is calculated by integrating the force over the displacement:

• Graphically, work is the area under the force vs. position curve between and .

• Example: If a force varies with , divide the path into segments, calculate work for each, and sum the results. As segments become infinitesimal, the sum becomes the integral.

Work Done by a Spring

The force required to compress or stretch a spring is proportional to the displacement:

• k is the spring constant, unique to each spring.

• The work done by the spring force as it moves from to is:

Power

Definition and Units

Power is the rate at which work is done or energy is transferred.

• SI Unit: Watt (W) = 1 joule/second

• US Customary Unit: Horsepower (hp), where 1 hp = 746 W

• 1 kilowatt-hour (kWh) = J

Examples and Applications

• Example: How long does it take a worker producing 200 W of power to do 10,000 J of work? s

• Comparison: If two people do the same amount of work over different times, the one who does it faster produces more power.

Worked Examples

Example: Block Pulled on a Frictionless Surface

• A 6.0-kg block is pulled by a 12 N force over 3.0 m. Find its final speed. m/s

Example: Block Pulled with Friction

• A 2.0-kg block is pulled by an 8 N force over 3.0 m with a coefficient of kinetic friction .

  • Friction force: N

  • Work by applied force: J

  • Work by friction: J

  • Net work: J

  • Final speed: m/s

Example: Crate Pulled Up an Incline

• 10.0-kg crate pulled 5.0 m up a 20° incline with initial speed 1.50 m/s, pulling force 100 N, kinetic friction 36.8 N.

  • Work by gravity: J

  • Work by friction: J

  • Work by applied force: J

  • Net work: J

  • Change in kinetic energy:

  • Final speed: m/s

Summary Table: Key Equations and Concepts

Additional info:

• All examples and applications are based on standard introductory physics problems.

• Units and conventions follow SI unless otherwise noted.