Work & Energy Lecture Notes

Work & Energy

Key Concepts

• Energy Conservation: The total energy of an isolated system remains constant. Energy can transform between forms (kinetic, potential, chemical, electrical, thermal, nuclear).
• Energy Transfer: Energy is transferred between objects or systems through work and heat. The unit for energy, work, and heat is the joule (J), where J = N \cdot m.
• Work Done by a Constant Force: The work done by a constant force F is given by W = Fd\cos\theta, where d is the distance moved and {\theta} is the angle between F and d. If {\theta} = 90^\circ, then W = 0.
  • Example: Pushing a trolley with F = 50 N at 30^\circ to the horizontal over a distance of d = 4 m. The work done is W = (50 \text{ N})(4 \text{ m})\cos(30^\circ) = 173 J.
• Physiological Work: Muscles under tension require energy to re-trigger muscle fiber contractions. This is termed physiological work. No physical work is done if there is no displacement, even if muscles ache from holding a heavy object.
• Efficiency:
  • Efficiency is the ratio of useful output (physical work) to metabolic energy used.
  • Formula: \text{Efficiency} = \frac{\text{work out}}{\text{energy used}}
  • Typical efficiency ranges from 5-20%, depending on the muscles used.

Kinetic Energy

• Definition: The kinetic energy (KE) of a moving object is given by KE = \frac{1}{2}mv^2, where m is the mass and v is the velocity.
  • Example: A 60 kg person sprinting at 9 m/s has a kinetic energy of KE = \frac{1}{2}(60 \text{ kg})(9 \text{ m/s})^2 = 2430 J.
• Work-Kinetic Energy Principle: The change in kinetic energy \Delta(KE) equals the work done on the object: \Delta(KE) = W = Fd.
  • Example: To find the average force required for a runner to gain a KE of 2430 J over 20 m: F = \frac{\Delta(KE)}{d} = \frac{2430 \text{ J}}{20 \text{ m}} = 121.5 N.

Work-Energy Principle

• Application: To find the constant net force F{\text{net}} required to accelerate a bus from speed v1 to v_2 over a distance d:
  • \Delta(KE) = \frac{1}{2} mv2^2 - \frac{1}{2} mv1^2
  • \Delta(KE) = F{\text{net}} d = \frac{1}{2} m(v2^2 - v_1^2)
  • F{\text{net}} = \frac{m(v2^2 - v_1^2)}{2d}

Gravitational Potential Energy

• Definition: The gravitational potential energy (GPE) of an object is given by GPE = mgy, where m is the mass, g is the acceleration due to gravity, and y is the height.
• Work Done: The work done by an external force F{\text{ext}} to raise an object of mass m through a height h is W = F{\text{ext}} d = mgh.
• Change in GPE: The change in gravitational potential energy is \Delta(GPE) = mgy2 - mgy1 = mgh.

Conservation of Energy

• Principle: In a closed system, the total energy (KE + GPE) remains constant. For example, if an object falls freely due to gravity, GPE decreases while KE increases, but their sum remains constant.

Power

• Definition: Power is the rate of transferring energy. It is measured in joules per second (J/s) or watts (W).
• Formula: P = \frac{\Delta \text{Energy}}{\Delta t}
  • Example: A 60 kg sprinter goes from 0 to 10 m/s and acquires 3000 J of kinetic energy in 4.0 s. The average power is P = \frac{3000 \text{ J}}{4.0 \text{ s}} = 750 \text{ W}.
• Human Power Output: Typical maximum human power output is around 500-900 W in activities like cycling or climbing stairs.

Power vs. Energy (Endurance)

• Limitations: Biomechanical processes are limited by either:
  • Power: The ability to deliver a lot of energy in a short time.
  • Total energy: The ability to deliver energy over a relatively long time.
• Examples:
  • Sprinting short distance: Power-limited.
  • Cycling uphill as fast as possible: Power-limited.
  • Lifting a very heavy barbell: Power-limited.
  • Light barbell lifted with 100 repetitions: Energy-limited.
  • Climbing up a mountain (2000 m gain in altitude): Energy-limited.