Frictional Work and Kinetic Energy

Work–Energy Principle

• Statement: \Delta K = Kf - Ki = W_{\text{net}}
  • K = \frac{1}{2}mv^2 is kinetic energy.
  • If the only non-conservative force doing work is kinetic friction, then W{\text{net}} = W{\text{friction}}.

Interpreting the Sign of \Delta K

• The transcript highlights that the final kinetic energy is zero (the object comes to rest).
• The initial kinetic energy is a positive value (object was moving).
• Therefore, \Delta K = Kf - Ki = 0 - Ki = -Ki < 0.
  • A negative \Delta K indicates an energy loss from the system.

Work Done by Kinetic Friction

• Friction does negative work because its force is opposite the direction of motion.
  • Formula: W{\text{friction}} = -fk\,d = -\mu_k N d.
• The negative sign ensures the work removes energy from the system, matching the negative \Delta K found above.

Numerical Reference ("six forty")

• The speaker mentions replacing something "by six-forty (640)"; contextually this is most likely:
  1. A force magnitude F = 640\,\text{N}, or
  2. A work/energy quantity W = 640\,\text{J}.
• Without additional lines, assume it is the magnitude plugged into the work or energy equation.
  • Example if it were the work: W_{\text{friction}} = -640\,\text{J}.
  • Then K_i = 640\,\text{J} so that \Delta K = -640\,\text{J}, bringing the object to rest.

Angle & Cosine Clarification

• Transcript quote: "It’s not the cosine of this; this is the cosine…"
  • Common pitfall: using the wrong angle when applying W = Fd\cos\theta.
  • Correct angle (\theta) is between the force vector and displacement vector.
  • For friction on a horizontal surface, \theta = 180^\circ, so \cos 180^\circ = -1.

Practical Implications

• Understanding sign conventions prevents errors in:
  • Computing energy changes.
  • Predicting whether a force adds or removes mechanical energy.
• Demonstrates why friction always converts mechanical energy into other forms (usually thermal).

Quick Check Questions

• What happens to the kinetic energy of a block sliding on a rough surface until rest?
• If friction does -500\,\text{J} of work, what was the block’s initial kinetic energy?
• Why is the cosine term -1 for kinetic friction on horizontal motion?