Work and Kinetic Energy: Chapter 6

• Work and Kinetic Energy is covered in Chapter 6 (12th edition), pp. 181-221.

Learning Objectives

• Define work.
• Identify and describe forms of work.
• Calculate the work done by a constant force.
• Define and explain the kinetic energy of a body.
• Apply work-energy principles/theorem to solve problems.
• Calculate the work done by a varying force.
• Define power and solve related problems.

Work Done by a Constant Force

• Work is done when a constant force acts on and displaces a body.
• For a constant force F acting on a particle, causing a straight-line displacement s at an angle φ with respect to the force, the work W done by the force on the body is given by: W = F · s = Fscosφ
• The SI unit of work is the joule (J), equivalent to a newton-meter (N·m): 1 J = 1 N·m
• Only the component of force parallel to the displacement does work on the car; F|| = Fcosφ
• The work done by a force is the product of the force times the parallel distance over which it acts , and it is a scalar quantity.

Positive Work

• Occurs when the angle φ between the force and displacement is between zero and 90 degrees.
• The force has a component in the direction of displacement.
• The work on the object is positive.
• W = F||s = (Fcosφ)s

Negative Work

• Occurs when the force has a component opposite to the direction of displacement.
• Corresponds to φ between 90° and 270°.
• Work done on the object is negative.
• The equation is W = F||s = (Fcosφ)s
• Mathematically, W < 0 because Fcosφ is negative.

Zero Work

• The force is perpendicular to the direction of displacement, φ= 90°.
• The force does no work on the object.
• A force acting on an object with a perpendicular component F⊥ to the object's displacement does no work.

No Work "Workout"

• The net work is zero if positive and negative work cancel each other, despite muscles being exercised.

Total Work

• Work done by several constant forces can be computed in two ways:
  • Method I: As the algebraic sum of the work done by the individual forces.
  • Method II: As the product of the displacement and the component of the net force in the direction of the displacement.

The Work-Energy Theorem

• The total work done on an object changes its position and speed.
• When a particle undergoes a displacement:
  • It speeds up if Wtot > 0.
  • Slows down if Wtot < 0.
  • Maintains the same speed if Wtot = 0.
• The equation is Wtot = K2 − K1 = ΔK

Physical Meaning of Kinetic Energy

• Based on the example, physical meanings of kinetic energy can be deduced:
  • From part 1, Wtot = K − 0 = K.
    • The kinetic energy of a particle equals the total work done to accelerate it from rest to its present speed.
    • This confirms the definition: K = 1/2 mv^2.
  • From part 2, kinetic energy of a particle is the total work the particle can do in being brought to rest.

Comparing Kinetic Energies

• Mass and velocity both affect the kinetic energy of a body.
  • Same mass, same speed, different directions of motion: same kinetic energy.
  • Twice the mass, same speed: twice the kinetic energy
  • Same mass, twice the speed: four times the kinetic energy.

Work Done by a Varying Force, Straight-Line Motion

• An example is driving a car, alternating between gas and brake.
• The effect is a variable positive or negative force of various magnitude along a straight line.
• The work done by the forces in the total displacement from x1 to x2 is approximately W= FaxΔxa + FbxΔxb + ⋯
  • In the limit, the sum becomes the integral of Fx from x1 to x2: ∫xFx dx (varying x-component of force, straight-line displacement)
• In the case where Fx is constant, the work done as a particle moves from x1 to x2, is given as: ∫xFx dx=Fx(x2-x1)

The Stretch of a Spring

• The force required to keep a spring stretched beyond its unstretched length by an amount x is given as Fx = kx, where k is the spring constant.
• The work done by this force when the elongation goes from zero to a maximum value X is W = ∫xFx dx = ∫xkx dx =1/2 kX^2
• Graphically, the work done is given as: W=1/2kX^2

Power

• Power is the time rate at which work is done.
• The average power is Pav =ΔW/ Δt
  • Instantaneous power is P = limΔt→0 ΔW/ Δt = dW/ dt
• In mechanics, power can also be expressed in terms of force and velocity.
  • ΔW = F|| Δs
  • Average power is Pav = F|| ( Δs/ Δt) = F Uav
  • Instantaneous power P is the limit of this expression as Δt → 0: P = F||U
• In terms of scalar product: P=F v
• The same work can be done in different situations, but the power, the rate at which work is done, will be different.