To determine Liz's minimum scale reading during the elevator trip, we need to analyze the forces acting on her as the elevator accelerates and decelerates. The scale reading reflects the normal force exerted on Liz, which changes depending on the elevator's motion. Let's break this down step by step.
Understanding Forces in an Elevator
When an elevator accelerates or decelerates, the apparent weight (or scale reading) of a person inside changes due to the effects of acceleration. The scale measures the normal force, which can be calculated using Newton's second law:
- Normal Force (N): This is what the scale reads.
- Weight (W): This is the gravitational force acting on Liz, calculated as W = mg, where m is mass and g is the acceleration due to gravity.
- Acceleration (a): This is the acceleration of the elevator, which can be positive (upward) or negative (downward).
Calculating Liz's Weight
First, we need to find Liz's mass. Given that her weight is 800 N, we can use the formula:
Weight (W) = mass (m) × g
Rearranging gives us:
m = W / g = 800 N / 10 m/s² = 80 kg
Scale Reading During Acceleration
When the elevator accelerates upwards, the normal force increases. The net force acting on Liz can be expressed as:
N - mg = ma
Rearranging gives:
N = m(g + a)
To find the minimum scale reading, we need to consider the maximum acceleration of the elevator. However, we need to determine the acceleration first.
Acceleration Between Floors
From the problem, we know that the elevator covers 6 meters between floors 2 and 5 in 1 second. Since there are three segments (2nd to 3rd, 3rd to 4th, and 4th to 5th), the total distance is:
Total Distance = 6 m × 3 = 18 m
Since the elevator covers this distance in 3 seconds (1 second per segment), we can find the average speed:
Average Speed = Total Distance / Time = 18 m / 3 s = 6 m/s
Using the formula for average speed during uniform acceleration:
Average Speed = (Initial Speed + Final Speed) / 2
Since the elevator starts from rest, the initial speed is 0. Thus:
6 m/s = (0 + v_f) / 2
Solving for final speed (v_f) gives:
v_f = 12 m/s
Finding Acceleration
Now, we can calculate the acceleration (a) using the formula:
a = (v_f - v_i) / t
Here, v_i = 0, v_f = 12 m/s, and t = 3 s:
a = (12 m/s - 0) / 3 s = 4 m/s²
Calculating the Minimum Scale Reading
Now that we have the acceleration, we can find the scale reading during the upward acceleration:
N = m(g + a) = 80 kg × (10 m/s² + 4 m/s²) = 80 kg × 14 m/s² = 1120 N
Scale Reading During Deceleration
When the elevator decelerates while moving downwards, the normal force decreases. The equation becomes:
N - mg = -ma
Rearranging gives:
N = m(g - a)
Using the same mass and the previously calculated acceleration:
N = 80 kg × (10 m/s² - 4 m/s²) = 80 kg × 6 m/s² = 480 N
Final Thoughts
Throughout the elevator trip, Liz's scale reading fluctuates based on the elevator's acceleration and deceleration. The minimum scale reading occurs when the elevator is decelerating, which is 480 N. Therefore, the minimum scale reading during the trip is:
480 N
