EXAM #3: DAY 12 Vertical Dynamics and Apparent Weight in Elevator Motion

Vertical Dynamics in Elevator Motion

Introduction to Vertical Dynamics

Vertical dynamics involves analyzing the forces and motion of objects moving along the vertical axis, such as elevators. Understanding these dynamics is essential for applying Newton's laws to real-world scenarios, especially when considering the concept of apparent weight.

Representing Motion in an Elevator

Stages of Elevator Motion

• Stage 1: At rest on the first floor (velocity and acceleration are zero).

• Stage 2: Accelerating upward (velocity increases, acceleration upward).

• Stage 3: Moving upward at constant speed (velocity constant, acceleration zero).

• Stage 4: Slowing down while moving upward (velocity upward, acceleration downward).

• Stage 5: At rest on the third floor (velocity and acceleration are zero).

Each stage can be represented with motion diagrams, velocity and acceleration vectors, and free-body diagrams to visualize the forces involved.

Free-Body Diagrams for Elevator Motion

Free-body diagrams (FBDs) are essential tools for visualizing the forces acting on the elevator or its passengers during each stage of motion. The main forces are the tension in the cable (or normal force from the floor) and the gravitational force (weight).

Newton's Laws Applied to Elevator Motion

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its acceleration:

$a = \frac{F_{\text{net}}}{m}$

Where $F_{\text{net}}$ is the vector sum of all forces acting on the object. The direction of acceleration matches the direction of the net force.

Kinematics in Elevator Problems

To analyze elevator motion, kinematic equations are used to relate velocity, acceleration, and displacement:

• $v_f = v_i + a \Delta t$

• $x_f = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$

• $v_f^2 = v_i^2 + 2a \Delta x$

Apparent Weight and Normal Force

Definition of Apparent Weight

Apparent weight is the normal force exerted by a surface (such as a scale) supporting an object. It is the force you "feel" as your weight, which can differ from your actual weight depending on acceleration.

• When accelerating upward, apparent weight increases.

• When accelerating downward, apparent weight decreases.

• At constant velocity or rest, apparent weight equals true weight.

Scales and Normal Force

Scales do not measure weight directly; they measure the normal force, which can change depending on the acceleration of the object or person on the scale.

Situation	Scale Reading (N)	Explanation
Object at rest	50	Normal force equals weight
Object pushed down	70	Normal force greater than weight
Object lifted up	0	Normal force less than weight (possibly zero)

Calculating Apparent Weight in an Elevator

The apparent weight $w_{\text{app}}$ of a person in an accelerating elevator is given by:

$w_{\text{app}} = m(g + a)$

• If $a > 0$ (upward acceleration), $w_{\text{app}} > mg$

• If $a < 0$ (downward acceleration), $w_{\text{app}} < mg$

• If $a = 0$, $w_{\text{app}} = mg$

Where $m$ is mass, $g$ is gravitational acceleration, and $a$ is the elevator's acceleration (positive upward).

Problem Solving: Elevator Example

Sample Problem

An elevator (with passenger) has a mass of 1,200 kg. It slows from 4 m/s to 0 m/s in 2 seconds. Find the tension in the cable during this time.

• Calculate acceleration: $a = \frac{v_f - v_i}{\Delta t} = \frac{0 - 4}{2} = -2$ m/s$^2$ (downward)

• Net force: $F_{\text{net}} = m a = 1200 \times (-2) = -2400$ N (downward)

• Weight: $W = mg = 1200 \times 9.8 = 11,760$ N (downward)

• Tension: $T = W + F_{\text{net}} = 11,760 + (-2400) = 9,360$ N

Conclusion: The tension is less than the weight when the elevator is slowing down while moving upward.

Conceptual Questions and Applications

When Does Apparent Weight Change?

• Heavier than normal: When the elevator accelerates upward.

• Lighter than normal: When the elevator accelerates downward.

• Normal weight: When the elevator is at rest or moving at constant velocity.

Graphical Representations

Velocity, acceleration, and force can be represented as functions of time for each stage of elevator motion. The scale reading (apparent weight) varies with acceleration.

Summary Table: Apparent Weight in Elevator Motion

Stage	Acceleration	Apparent Weight
At rest / constant speed	0	Equal to true weight
Speeding up (upward)	Upward	Greater than true weight
Slowing down (upward)	Downward	Less than true weight
Speeding up (downward)	Downward	Less than true weight
Slowing down (downward)	Upward	Greater than true weight

Key Takeaways

• Apparent weight is determined by the normal force, not just gravity.

• Elevator problems require careful identification of forces and application of Newton's laws.

• Scales measure the normal force, which can differ from true weight during acceleration.