Overview
This lecture covers how to calculate the work done by the gravitational force, analyzes different cases such as rising or falling objects, and examines the roles of applied forces and gravity with real-world examples.
Work Done by Gravitational Force
- Work is calculated using (W = F \cdot d \cdot \cos(\theta)), with gravitational force (F = mg).
- For a rising object, displacement is upward, gravity acts downward ((\theta = 180^\circ)), so work by gravity is negative: (W = -mgd).
- For a falling object, displacement and gravity are both downward ((\theta = 0^\circ)), so work by gravity is positive: (W = +mgd).
- Negative work by gravity indicates loss of kinetic energy; positive work indicates a gain in kinetic energy.
Work by Applied Forces and Gravity
- When lifting or lowering an object, an upward applied force counters gravity.
- Net work ((W_{net})) equals change in kinetic energy: (K_f - K_i = W_{applied} + W_{gravity}).
- If initial and final kinetic energies are zero (object starts and ends at rest), (W_{applied} = -W_{gravity}).
- The equation for applied work depends on the direction between gravity and displacement, not the applied force.
- The work done by gravity is path-independent; lifting straight up or along a path gives the same result._
Examples: Lifting and Lowering Objects
- When moving up, the applied force does positive work, gravity does negative work.
- When moving down, gravity does positive work, applied force does negative work.
- The magnitudes of work by gravity and the applied force are only equal if kinetic energy does not change.
Elevator Example
- Forces on a descending elevator: gravity (downward) and tension in the cable (upward).
- Displacement is downward; gravity does positive work, tension does negative work.
- If analyzing a passenger, substitute the normal force for tension.
Key Terms & Definitions
- Work (W) — The product of force, displacement, and the cosine of the angle between them: (W = Fd\cos\theta).
- Gravitational Force (mg) — The force exerted by gravity on an object.
- Applied Force ((F_a)) — The upward force applied to lift or support an object against gravity.
- Kinetic Energy (K) — The energy of motion, given by (K = \frac{1}{2}mv^2).
Action Items / Next Steps
- Practice identifying forces and work directions in everyday scenarios.
- Review examples involving lifting and lowering objects for path independence.
- Complete assigned textbook problems on work and kinetic energy changes.