Mass v. Weight and Vertical Motion

Weight, Mass, and the Dreaded Elevator Problem

• Weight: The force of gravity acting on an object with mass.

  • Weight causes objects to accelerate at ( g = 9.81 \text{ m/s}^2 ) towards the center of the Earth.

  • Units of weight: Newtons (N).

Weight: True or False?

• Journal Exercise: Evaluate true or false for the following statements:

  • The mass of an object depends on its location: False - Mass is invariant regardless of location.

  • The weight of an object depends on its location: True - Weight varies with the gravitational force in that location.

  • Mass and weight are the same but with different units: False - Mass measures matter (kg), while weight measures force (N).

Weight vs. Mass

• Weight is the gravitational force acting on an object:

  • Calculated using the equation: ( W = m \cdot g )

  • Always directed downwards towards Earth's center.

Apparent Weight

• Apparent weight: The weight observed when acceleration is present.

  • Example: Standing on a scale in an elevator—the scale reads your apparent weight.

Apparent Weight Practice

• Cases analyzing apparent weight changes:

  • Ascending and speeding up: Apparent weight > true weight.

  • Descending and speeding up: Apparent weight < true weight.

  • Ascending at constant speed: Apparent weight = true weight.

  • Ascending and slowing down: Apparent weight < true weight.

  • Descending and slowing down: Apparent weight > true weight.

  • Descending at constant speed: Apparent weight = true weight.

Constant Vertical Velocity Example

• Example: Leaf falling at terminal velocity

  • At terminal velocity, acceleration is zero, indicating weight force equals the drag force.

Calculating Apparent Weight

• Apparent weight calculated via net force:

  • When at rest, forces acting: gravitational force (weight) and normal force (scale force).

  • Net Force: 0 N (static equilibrium).

Free-body Diagram and Vector Equation

• Free-body diagram shows forces acting on the object.

• Vector equation: ( F_{scale} = W_{apparent} ) and ( W_{g} = m \cdot g )

Equilibrium and True Weight

• In equilibrium, the scale shows true weight:

  • Example calculation for mass of 65.0 kg:

  • Weight = ( W = 65.0 ext{ kg} \cdot 9.81 ext{ m/s}^2 ) = 637 N.

Accelerating Upwards Example

• Example scenario: Crate lifted by a rope.

  • Forces in positive direction analyzed:

  • Net force involves both the gravitational and tension forces in the rope.

Accelerating Elevator

• When an elevator accelerates:

  • Need to account for the acceleration effect on apparent weight.

  • Example: If elevator accelerates upwards at ( 2.00 ext{ m/s}^2 ) using upward direction as positive.

Free-body Diagram and Notifications

• Diagram should show the forces on an object when the elevator moves.

  • Include direction of net force.

Equations Under Non-equilibrium

• Since the situation is not in equilibrium, derive apparent weight equations including net forces.

Accelerating Downwards Example

• Scenario: Sky diver in free fall.

  • Analysis of upward and downward forces in motion.

  • Net forces determined through vector dynamics regarding gravity's effect on apparent weight.

Another Accelerating Elevator

• New scenario: Elevator accelerating downwards at ( 2.00 ext{ m/s}^2 ).

  • Another free-body diagram and calculation of net force.

Writing Vector Equations

• The net force is directed downwards, indicating acceleration and apparent weight are both negative.

Problem Set: Determine Apparent Weight

• Calculate the apparent weight of a 67 kg man in different elevator conditions:

  • At rest: apparent weight = true weight.

  • Ascending, speeding up at 1.5 m/s²: apparent weight > true weight.

  • Ascending, slowing down at -1.2 m/s²: apparent weight < true weight.