Study Guide on Work, Energy, and Power

Work and Energy

Introduction to Work and Energy

• Work and Energy as tools for solving problems involving forces and motion.
• Energy and momentum are conserved quantities.
• Conservation laws are valuable, especially for systems with multiple objects where detailed force analysis may be impractical.
• Applicable to various phenomena, including atomic and subatomic levels where Newton's laws may not hold.
• Work and Energy are scalar quantities; they do not have direction.

Work Done by a Constant Force

• The work done on an object by a constant force is defined as the product of:
  • The magnitude of the displacement (
  • The component of the force parallel to the displacement.
• Mathematically, the work done ( can be expressed as: $W = F imes d imes ext{cos}( heta)$ where:
  • F = Magnitude of the constant force
  • d = Magnitude of the displacement
  • θ = Angle between the direction of the force and displacement

Special Cases of Work Done

1. Force and motion in the same direction (θ = 0°):

   • e.g., Pushing a cart horizontally with a force of 30 N over a distance of 10 m yields:
     $W = 30 imes 10 imes ext{cos}(0) = 300 ext{ J}$

2. Force and motion are perpendicular (θ = 90°):

   • Example: Holding a bag of groceries while walking horizontally requires no horizontal force, hence no work is done.

3. Force and motion are opposite (θ = 180°):

   • e.g., Friction opposes motion, therefore:
     $W = F imes d imes ext{cos}(180) = -F imes d$
   • This indicates that negative work is done by friction.

Clarification on Work Calculation

• Important to state:
  • Whether work is done by a specific object or on it.
  • Whether the work is calculated from a particular force or net force.

Examples of Work Done

1. Example 1: A person pulls a 50-kg crate 40 m along a rough floor at a 37° angle.

   • Find work done by each force acting on the crate and net work done on the crate.

2. Work Done by Gravity should be zero when forces are at right angles to displacement (normal and gravitational force).

   • $W_{ ext{gravity}} = 0$

3. Example 2 (Backpack Work): Calculate work required to carry a 15.0-kg backpack up a hill, the work done by gravity,
   and net work done at a constant velocity.

Does Earth do work on the Moon?

• Analyze the nature of the Moon's orbit around Earth:
  • Does gravity do positive work, negative work, or no work on the moon?

Work Done by a Varying Force

• As a rocket moves away from Earth, work done against varying gravitational force, which decreases with distance.
• Work done by a varying force can be estimated graphically by plotting against distance:
  1. Plot force component parallel to motion against distance.
  2. Divide distance into segments (e.g., Dd).
  3. Compute work done for each segment as the area of rectangles.
  4. Total work is the sum of these areas. In the limit of Dd approaching zero, it equals the area under the curve.

Work-Energy Theorem

Definition and Concepts

• Define energy as the ability to do work.
• Kinetic energy (KE) is energy due to motion: KE = rac{1}{2}mv^2 where:
  • m = mass
  • v = velocity.
• The work done on an object equals the change in its kinetic energy,
  $W = ext{ΔKE} = KE_f - KE_i$.

Key Points on Kinetic Energy

• If positive net work is done, the object's KE increases.
• If negative net work is done, the object's KE decreases.
• If no net work is done, the object's KE stays constant.
• Doubling the mass while keeping speed constant doubles KE.
• Doubling speed while keeping mass constant quadruples KE.

Example Calculations

1. Baseball:

   • A 145-g baseball at 25 m/s,
     • a) Calculate KE.
     • b) Determine net work done to reach this speed, starting from rest.

2. Car Acceleration:

   • Find net work needed to accelerate 1000 kg car from 20 m/s to 30 m/s.

Conceptual Example: Braking Distance

• A car traveling at 60 km/h stops in 20 m,
  • If speed doubled: The stopping distance triples,
  • This is due to KE being proportional to the square of speed.

Potential Energy - Gravitational and Elastic

Gravitational Potential Energy (GPE)

• Defined as the energy stored due to an object's position in a gravitational field: $GPE = mgh$ where:
  • m = mass,
  • g = acceleration due to gravity (9.8 m/s²),
  • h = height above a reference level.

Calculating Changes in Potential Energy

• Changes in GPE when objects move:
  • Work done by an external force to raise mass effectively matches the change in potential energy.
  • Work done by gravity results in a decrease in potential energy.

Elastic Potential Energy

• Associated with the compression or stretching of a spring: PE = rac{1}{2}kx^2 ,
  • k = spring constant,
  • x = displacement from equilibrium position.

Hooke's Law

• The force exerted by a compressed spring is proportional to the displacement (
  , acting to restore to its natural length,
  $F = -kx$

Conservation of Energy

• Conservative Forces: Work done only depends on initial and final positions.
• Non-conservative Forces: Work depends on the path taken (e.g., friction).

Conservation Principle Application

• Mechanical energy conservation example: Free fall of a rock:
  • Initially all potential, converting to kinetic as it falls.

Power

Definition

• Power is the rate at which work is done or energy is transformed: P = rac{W}{t}
  • W = work done,
  • t = time.
• Unit of power is the watt (W):
  • 1 W = 1 J/s = 1 horsepower (hp) = 746 W.

Example Calculations

1. Jogger Power Output:

   • A 60 kg jogger runs up stairs of height 4.5 m in 4.0 s:
     • Calculate average power output.

2. Automobile Power Requirements:

   • Power calculations for a car climbing hills or accelerating in terms of resistance.

Efficiency

• Defined as the ratio of useful work output to total work input: Efficiency () = rac{W_{out}}{W_{in}}
  • Usually expressed as a percentage. Efficiency values are always less than one due to energy losses (e.g., to heat).

Example Efficiency Calculations

• Calculate the time taken to lift an object using a motor with a known efficiency and power input.