Study Notes on Work and Kinetic Energy

Chapter 7: Work and Kinetic Energy

Introduction

This chapter discusses the concepts of work and kinetic energy, including definitions, equations, and applications that will be explored in detail throughout the section.

Forms of Energy

• Mechanical Energy: Focus point for this chapter.

  • Kinetic Energy: Energy associated with motion.

  • Potential Energy: Energy associated with position.

• Chemical Energy: Involves energy stored in chemical bonds.

• Electromagnetic Energy: Energy propagated through electromagnetic waves.

• Nuclear Energy: Energy stored in the nucleus of atoms.

Work

• Definition: Work is the transfer of energy via force applied over a distance.

• Mathematical Expression: W=\left(F\cos\theta\right)\Delta x where:

  • F: Magnitude of the force applied,

  • Δx: Magnitude of the object's displacement,

  • θ: Angle between \overrightarrow{F} and \Delta\overrightarrow{x}

• Properties:

  • Work is a scalar quantity.

  • It provides a link between force and energy.

• Units:

  • SI / MKS Unit: Newton • meter = Joule

  • 1J=N\cdot m = m² / s²

  • US Customary Unit: foot • pound = \operatorname{ft}\cdot\operatorname{lb}

More About Work

• Conditions of Work Done:

  • The work done by a force is zero when the force is perpendicular to the displacement.

  • Example: cos(90°) = 0.

  • For multiple forces acting on an object, the total work done is the algebraic sum of the work done by each force.

• Work Significance:

  • Work can be positive or negative:

  • Positive Work: When force and displacement are in the same direction.

  • Negative Work: When force and displacement are in the opposite directions.

• Net Work Done: If multiple forces are present,
  W_{net}=F_{net,x}\cdot\Delta x

  Where:
  F_{net,x}=F_{1x}+F_{2x}+\ldots+F_{nx}

Kinetic Energy and Work-Energy Theorem

• Recall: The work done by a constant force can be expressed as:
  W_{net}=\frac12mv^2-\frac12mv_0^2

  • Terms Explained:

  • v_0: Initial velocity

  • v: Final velocity

  • Using Newton's 2nd Law:
    F_{net,x}=ma_{x}

  • And the 1D kinematic equation:
    \Delta x=\frac{v_{x^{}}^2-v_{0x}^2}{2a_{x}}

  • The relation:
    W_{net}=F_{net,x}\cdot\Delta x yields:
    W_{net}=\left(ma_{x}\right)\left(\frac{v_{x}^2-v_{0x}^2}{2a_{x}}\right)=\frac12m\left(v_{x}^2-v_{0x}^2\right)

Kinetic Energy

• Expression: The kinetic energy (K) of an object with mass (m) and speed (v) is given by:
  K=\frac12mv^2

Implications of Work:

• Positive Net Work: Increases speed (and kinetic energy).

• Negative Net Work: Decreases speed (and kinetic energy).

• Work-Energy Theorem: W_{net}=K-K_0=\Delta K

  • Represents the work done as a change in kinetic energy.

Work and Kinetic Energy

• An object's kinetic energy can be thought of as the amount of work the moving object could do while coming to rest.

  • Example: A moving hammer possesses kinetic energy and is capable of doing work on a nail.

A Variable Force: The Spring

• When dealing with variable forces, such as those exerted by a spring, the force exerted is proportional to the displacement (Δx) from its equilibrium position:

  • Hooke's Law: F_x = kx

  • where k is the spring constant.

  • SI Unit of k: N/m.

Work Done by Variable Force: Spring Force

• The work done by a variable force, such as the spring force, is given by:
  W=\frac12kx^2

  • This work results in energy being stored in the spring.

• The area under the force (F) vs. displacement (x) graph represents the work done.

Power

• Definition: Power is the rate at which energy transfer takes place.

• Mathematical Expression: \overline{P}=\frac{W}{t}=F\overline{v}

  • Where P is power, W is work done, and t is the time period.

• Units:

  • SI Unit: Watts (W)
    W=\frac{J}{s}=\frac{\operatorname{kg}\cdot m^2}{s^2}

  • US Customary Unit: Horsepower (hp)

• Conversion Factor: 1hp=550\frac{\operatorname{ft}\cdot\operatorname{lb}}{s}=746W

  • Notably, a kilowatt-hour (kWh) used in electric bills is another measurement related to power and energy transfer.